Research Article Open Access
The Production of Bio-energy and its Properties of Transport in the Living systems
Pang Xiao- feng*
Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of Chengdu 610054,China.
*Corresponding authors address: Pang Xiao-feng, Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of Chengdu 610054,China; E-mail: pangxf2006@aliyun.com
Received: May 5, 2017; Accepted: May13, 2017; Published: July 03, 2017
Citation: Pang Xiao- feng (2017) The Production of Bio-energy and its Properties of Transport in the Living systems.Int Struct Comput Biol 1(1):1-21.
Abstract
We here introduced the form of bio-energy in living system and eluciduted again the new theory of bio-energy transport along protein molecules in living systems based on the changes of structure and conformation of molecules arising from the energy, which is released by hydrolysis of adenosine triphosphate (ATP). In this theory, the Davydov’s Hamiltonian and wave function of the systems are simultaneously improved and extended. A new interaction have been added into the original Hamiltonian. The original wave function of the excitation state of single particles have been replaced by a new wave function of two-quanta quasicoherent state. In such a case, bio-energy is carried and transported by the new soliton along protein molecular chains. The soliton is formed through self- trapping of two excitons interacting amino acid residues. The exciton is generated by vibrations of amide-I (C=O stretching) arising from the energy of hydrolysis of ATP. The properties of the soliton are extensively studied by analytical method and its lifetime for a wide ranges of parameter values relevant to protein molecules is calculated using the nonlinear quantum perturbation theory. The lifetime of the new soliton at the biological temperature 300 K is enough large and belongs to the order of 10- 10 second orτ/τ0≥700. The different properties of the new soliton are further studied. The results show that the new soliton in the new model is a better carrier of bio-energy transport and it can play an important role in biological processes. This model is a candidate of the bio-energy transport mechanism in protein molecules.

Keywords: Form; Living system; Bio-energy; protein; Biological energy; Soliton; ATP hydrolysis; Amide; Exciton; Life time; Amino acid; Quasi-coherent state.
I. The Phosphorylation and de-Phosphorylation reactions in the Cell and the features of energy released in hydrolysis of ATP molecules
As it is known, Kal’kar first proposed the idea of aerobic phosphorylation, which is carried out by the phosphorylation coupled to the respiration. Belitser studied in detail the stoichometric ratios between the conjugated bound phosphate and the absorption of oxygen and gave further the ratio of the number of ionorganic phosphate molecules to the number of oxygen atoms absorbed during the respiration, which is not less than two. He thought also that the transfer of electrons from the substrate to the oxygen is a possible source of energy for the formation of two or more ATP molecules per atom of absorbed oxygen. Therefore Belitser and Kal’kar’s research results are foundations establishing modern theory of oxidative phosphorylation of ATP molecules in the cell [1-3].

In such a case we must know clearly the mechanism and properties of the oxidation process, which involves the transfer of hydrogen atoms from the oxidized molecule to another molecule, in while there are always protons present in water and in the aqueous medium of the cell, thus we may only consider the transfer of electrons in this process. The necessary number of protons to form hydrogen atoms is taken from the aqueous medium. The oxidation reaction is usually preceded inside the cell under the action of special enzymes, in which two electrons are transferred from the food substance to some kind of initial acceptor, another enzymes transfer them further along the electron transfer chain to the second acceptor etc. Thus a water molecule is formed in which each oxygen atom requires two electrons and two protons.

The main initial acceptors of electrons in cells are the oxidized forms NAD+ and NADP+ of NAD (nicotine amide adenine dinucleotide or pyridine nucleotide with two phosphate groups) molecules and NADP(nicotine amide adenine nucleotide phosphate or pyridine nucleotide with three phosphate groups) as well as FAD (flavin adenine dinucleotide or flavoquinone) and FMN (flavin mononucleotide).The above oxidized forms of these molecules serve for primary acceptors of electrons and hydrogen atoms through attaching two hydrogen atoms [3], which is expressed by NAD P + +2 H + +2 e NADPH+ H + MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadg eacaWGebGaamiuamaaCaaaleqabaGaey4kaScaaOGaey4kaSIaaGOm aiaadIeadaahaaWcbeqaaiabgUcaRaaakiabgUcaRiaaikdacaWGLb WaaWbaaSqabeaacqGHsislaaGccqGHsgIRcaWGobGaamyqaiaadsea caWGqbGaeyyXICTaamisaiabgUcaRiaadIeadaahaaWcbeqaaiabgU caRaaaaaa@4C6E@ Where NADP molecule becomes the reduced molecule NADPH. The NAD molecule has also the same active center as the NADP molecule; it can be converted to the reduced molecule NAD. H under combining with two atoms of hydrogen according to the reaction [3]: NA D + +2 H + +2 e NAD P + + H + MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadg eacaWGebWaaWbaaSqabeaacqGHRaWkaaGccqGHRaWkcaaIYaGaamis amaaCaaaleqabaGaey4kaScaaOGaey4kaSIaaGOmaiaadwgadaahaa WcbeqaaiabgkHiTaaakiabgkziUkaad6eacaWGbbGaamiraiaadcfa daahaaWcbeqaaiabgUcaRaaakiabgUcaRiaadIeadaahaaWcbeqaai abgUcaRaaaaaa@499B@ The NAD+ and NADP+ are the enzymes, which can perform the reaction of dehydrogenation on compounds containing the group of atoms through removing two hydrogen atoms.

In the presence of enzymes, such as pyridine-dependent hydrogenases, and with the participation NAD+ and NADP+ molecules two hydrogen atoms, including two protons and two electrons, are removed from this group of atoms. One proton and two electrons combine with the NAD+ and NADP+ molecule converting them to the reduced forms NADP . H or NAD . H and the second proton is released. This mechanism can be also used to oxidize lactic acid (lactate) with the formation of pyruvic acid (pyruvate) and NAD H, in which the reduced molecules NADP . H and NAD . H serve as electron donors (reducing agents) in other reactions. They are involved in a large number of biosynthetic processes, such as in the synthesis of fatty acids and cholesterol.

Therefore, the molecule NAD • H can serve as an electron donor in the process of oxidative phosphorylation, then the phosphorylation reaction is of [3] H + +NADH+3 H 3 P O 4 +3ADP+1/2 O 2 NA D + +4 H 2 O+3ATP MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaCa aaleqabaGaey4kaScaaOGaey4kaSIaamOtaiaadgeacaWGebGaeyyX ICTaamisaiabgUcaRiaaiodacaWGibWaaSbaaSqaaiaaiodaaeqaaO Gaamiuaiaad+eadaWgaaWcbaGaaGinaaqabaGccqGHRaWkcaaIZaGa amyqaiaadseacaWGqbGaey4kaSIaaGymaiaac+cacaaIYaGaam4tam aaBaaaleaacaaIYaaabeaakiabgkziUkaad6eacaWGbbGaamiramaa CaaaleqabaGaey4kaScaaOGaey4kaSIaaGinaiaadIeadaWgaaWcba GaaGOmaaqabaGccaWGpbGaey4kaSIaaG4maiaadgeacaWGubGaamiu aaaa@5AB5@ Where ADP is called the adenosine diphosphate. The abbreviated form of this reaction can be written as ADP+ P i ATP+ H 2 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaads eacaWGqbGaey4kaSIaamiuamaaBaaaleaacaWGPbaabeaakiabgkzi UkaadgeacaWGubGaamiuaiabgUcaRiaadIeadaWgaaWcbaGaaGOmaa qabaGccaaIWaaaaa@42F0@ Thus three ATP molecules are formed in the reaction, in which the synthesis of ATP molecule are carried out through the transfer of two electrons from the NAD • H molecule along the electron transport chain to the oxygen molecule in the mitochondria. In this way the energy of each electron is reduced by 1.14 eV. The reaction is called the phosphorylation of ADP molecules.

However, an ATP molecule can reacts with water in an aqueous medium, which results in the energy release of about 0.43eV under normal physiological conditions by virtue of some special enzymes. The reaction can be represented by AT P 4 + H 2 OAD P 3 +HP O 4 2 + H + +0.43eV MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaads facaWGqbWaaWbaaSqabeaacaaI0aGaeyOeI0caaOGaey4kaSIaamis amaaBaaaleaacaaIYaaabeaakiaad+eacqGHsgIRcaWGbbGaamirai aadcfadaahaaWcbeqaaiaaiodacqGHsislaaGccqGHRaWkcaWGibGa amiuaiaad+eadaqhaaWcbaGaaGinaaqaaiaaikdacqGHsislaaGccq GHRaWkcaWGibWaaWbaaSqabeaacqGHRaWkaaGccqGHRaWkcaaIWaGa aiOlaiaaisdacaaIZaGaamyzaiaadAfaaaa@523E@ Its abbreviated form is of ATP+ H 2 OADP+ P i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaads facaWGqbGaey4kaSIaamisamaaBaaaleaacaaIYaaabeaakiaad+ea cqGHsgIRcaWGbbGaamiraiaadcfacqGHRaWkcaWGqbWaaSbaaSqaai aadMgaaeqaaaaa@4300@ In this process ATP molecules are transformed as ADP molecules and the bio-energy of about 0.43eV is also released. Then it is referred to as de-phosphorylation reaction of ATP molecules.

We know from the above representations that an increase in free energy ΔG MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaae 4raaaa@3825@ in reaction and its decrease in reaction depend on their temperatures, concentrations of the ions Mg2+ and Ca2+ and on the pH value of the medium. Under the standard conditions Δ G 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam 4ramaaBaaaleaacaaIWaaabeaaaaa@390D@ =0.32 eV ( ~7.3 kcal/mole). If the appropriate corrections are made taking into consideration the physiological pH values and the concentration of Mg2+ and Ca2* inside the cell as well as the normal values for the concentrations of ATP and ADP molecules and inorganic phosphate in the cytoplasm we can obtain a value of ~0.54eV ( ~12.5 kcal/mole) for the free energy in the hydrolysis of ATP molecule. Hence the free energy for the hydrolysis of ATP molecules is not constant. But it is impossibly the same at different sites of the same cell if these sites have different concentrations of ATP, ADP, Pi, Mg2+, Ca2+.

On the other hand, cells contain a number of phosphorylated compounds the hydrolysis of which in the cytoplasm is associated with the release of free energy. Then the values for the standard free energy of hydrolysis for some of these compounds are also different.

The enzymes carrying out the above synthesis of ATP molecules from ADP molecules and inorganic phosphate in the coupling membranes of mitochondria are the same as in the cytoplasmic membranes of bacteria, which are mainly composed of F and F, which are joined to each other by the small proteins F5 and F6 . These proteins form the F - F complex or the enzyme ATPase, in which F is composed of five protein subunits and has the shape of a sphere with a diameter of about 9nm which projects above the surface of the membrane in the form of a protuberance. In the coupling membrane of mitochondria and the cytoplasmic membrane of bacteria the complex F - F is positioned so that the enzyme F is on the inside of the membrane [1-3].

The enzyme F can extend from one side of the membrane to the other and has a channel which lets protons through. When two protons pass through the complex F - F in the coupling mitochondrial membrane one ATP molecule is synthesized inside the matrix from an ADP molecule and inorganic phosphate. This reaction is reversible. Under certain condition the enzyme transports protons from the matrix to the outside using the energy of dissociation of ATP molecules, which may be observed in a solution containing isolated molecules of enzyme F and ATP. The largest two proteins in F, which is composed of five protein molecules, take part in the synthesis and dissociation of ATP molecules, the other three are apparently inhibitors controlling these reactions.

After removing enzyme F molecules from mitochondria the remaining F enzymes increase greatly the permeability of protons in the coupling membranes, which confirms that the enzyme F has really a channel for the passage of protons which is constructed by the enzyme F. However, the complete mechanism for the synthesis of ATP molecules by the enzyme ATP-ase is still not clearly known up to now.
II. The physical and biological foundations of construction of new theory
As it is known, many biological processes, such as muscle contraction, DNA reduplication, neuroelectric pulse transfer on the neurolemma and work of calcium pump and sodium pump, and so on, are associated with bioenergy transport through protein molecules, where the energy is released by the hydrolysis of adenosine triphosphate (ATP) in the living systems. Thus there here are always biological processes of energy transport from production place to absorption place in the living systems. In general, the bioenergy transport is carried out by virtue of protein molecules. Therefore, the study of the bioenergy transport along protein molecules is a very interesting subject in biology and has important significance in life science. However, understanding the mechanism of bioenergy transport in biomacromolecular systems has been a long-standing problem that remains of great interest today. As an alternative to electronic mechanisms [1], one can assume that the energy is stored as vibrational energy in the C=0 stretching mode (amide-I) of a protein molecular chain of polypeptide. Following Davydov’s idea [2], ones take into account the coupling between the amide-I vibrarional quantum (exciton ) and the acoustic phonon (molecular displacements) in the amino acid residues; Through the coupling, nonlinear interaction appears in the motion of the vibrartional quanta, which could lead to a self-trapped state of the vibrational quantum. The latter plus the deformational amino acid lattice together can travel over macroscopic distances along the molecular chains, retaining the wave shape, energy, momentum and other properties of the quasiparticle. In this way, the bioenergy can be transported as a localized “wave packet” or soliton. This is just the Davydov’s model of bioenergy transport in proteins, which was proposed in the 1970s [2,3].

Davydov model of bioenergy transport work at α − helical proteins as shown in Figure.1.
Figure 1: Structure of α − helical protein
Following Davydov idea [3], the Hamiltonian describing such system has in the form of H D = n [ ε 0 B n + B n J( B n + B n1 + B n B n+1 + ) ]+ n [ P n 2 2M + 1 2 w ( u n u n1 ) 2 ] + n [ χ 1 ( u n+1 u n1 ) B n + B n ]= H ex + H ph + H int                                           (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGib WaaSbaaSqaaiaadseaaeqaaOGaeyypa0ZaaabuaeaadaWadaqaaiab ew7aLnaaBaaaleaacaaIWaaabeaakiaadkeadaqhaaWcbaGaamOBaa qaaiabgUcaRaaakiaadkeadaWgaaWcbaGaamOBaaqabaGccqGHsisl caWGkbGaaiikaiaadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaaki aadkeadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaey4kaSIa amOqamaaBaaaleaacaWGUbaabeaakiaadkeadaqhaaWcbaGaamOBai abgUcaRiaaigdaaeaacqGHRaWkaaGccaGGPaaacaGLBbGaayzxaaGa ey4kaScaleaacaWGUbaabeqdcqGHris5aOWaaabuaeaadaWadaqaam aalaaabaGaamiuamaaDaaaleaacaWGUbaabaGaaGOmaaaaaOqaaiaa ikdacaWGnbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaam 4DaiaacIcacaWG1bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyD amaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaGGPaWaaWbaaS qabeaacaaIYaaaaaGccaGLBbGaayzxaaaaleaacaWGUbaabeqdcqGH ris5aaGcbaGaey4kaSYaaabuaeaadaWadaqaaiabeE8aJnaaBaaale aacaaIXaaabeaakiaacIcacaWG1bWaaSbaaSqaaiaad6gacqGHRaWk caaIXaaabeaakiabgkHiTiaadwhadaWgaaWcbaGaamOBaiabgkHiTi aaigdaaeqaaOGaaiykaiaadkeadaqhaaWcbaGaamOBaaqaaiabgUca RaaakiaadkeadaWgaaWcbaGaamOBaaqabaaakiaawUfacaGLDbaacq GH9aqpcaWGibWaaSbaaSqaaiaadwgacaWG4baabeaakiabgUcaRiaa dIeadaWgaaWcbaGaamiCaiaadIgaaeqaaOGaey4kaSIaamisamaaBa aaleaaciGGPbGaaiOBaiaacshaaeqaaaqaaiaad6gaaeqaniabggHi LdGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGOaGaaeymaiaabMcaaaaa@AD2E@ where ε 0 =0.205ev MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIYaGaaGim aiaaiwdacaqGLbGaaeODaaaa@3F14@ is is the amide-I quantum energy, -J is the dipole-dipole interaction energy between neighboring sites, B n + ( B n ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaDa aaleaacaWGUbaabaGaey4kaScaaOGaaiikaiaadkeadaWgaaWcbaGa amOBaaqabaGccaGGPaaaaa@3C11@ is the creation (annihilation) operator for an amide-I quantum excitation (exciton) in the site n, un is the displacement operator of amino acid residues at site n, Pn is its conjugate momentum operator, M is the mass of an amino acid molecule, w is the elasticity constant of the protein molecular chains, and χ 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaaaa@3893@ is an nonlinear coupling parameter and represents the coupling size of the exciton-phonon interaction. The wave function of the systems proposed by Davydov is in the form of | D 2 (t) >=| φ D (t)>1β(t)>= n φ n (t) Β n + exp( i n [ β n (t) Ρ n π n (t) u n ] )10>(2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaWaaqqaaeaaca WGebWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaacaGL hWoacqGH+aGpcqGH9aqpcaGG8bGaeqOXdO2aaSbaaSqaaiaadseaae qaaOGaaiikaiaadshacaGGPaGaeyOpa4JaaGymaiabek7aIjaacIca caWG0bGaaiykaiabg6da+iabg2da9maaqafabaGaeqOXdO2aaSbaaS qaaiaad6gaaeqaaaqaaiaad6gaaeqaniabggHiLdGccaGGOaGaamiD aiaacMcacqqHsoGqdaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiGacw gacaGG4bGaaiiCamaabmaabaGaeyOeI0YaaSaaaeaacaWGPbaabaGa eS4dHGgaamaaqafabaGaai4waiabek7aInaaBaaaleaacaWGUbaabe aakiaacIcacaWG0bGaaiykaiabfg6asnaaBaaaleaacaWGUbaabeaa kiabgkHiTiabec8aWnaaBaaaleaacaWGUbaabeaakiaacIcacaWG0b GaaiykaiaabwhadaWgaaWcbaGaamOBaaqabaGccaGGDbaaleaacaWG UbaabeqdcqGHris5aaGccaGLOaGaayzkaaGaaGymaiaaicdacqGH+a GpcaGGOaGaaGOmaiaacMcaaaa@781E@ | D 1 (t)>= n { φ n (t) B n + exp ( q [ α nq (t) a q + α nq * (t) a n ] ) }10 >          (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaads eadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcacqGH+aGp cqGH9aqpdaaeqbqaamaaceaabaGaeqOXdO2aaSbaaSqaaiaad6gaae qaaOGaaiikaiaadshacaGGPaGaamOqamaaDaaaleaacaWGUbaabaGa ey4kaScaaOGaciyzaiaacIhacaGGWbWaaiGaaeaadaqadaqaamaaqa fabaWaamWaaeaacqaHXoqydaWgaaWcbaGaamOBaiaadghaaeqaaOGa aiikaiaadshacaGGPaGaamyyamaaDaaaleaacaWGXbaabaGaey4kaS caaOGaeyOeI0IaeqySde2aa0baaSqaaiaad6gacaWGXbaabaGaaiOk aaaakiaacIcacaWG0bGaaiykaiaadggadaWgaaWcbaGaamOBaaqaba aakiaawUfacaGLDbaaaSqaaiaadghaaeqaniabggHiLdaakiaawIca caGLPaaaaiaaw2haaiaaigdacaaIWaaacaGL7baaaSqaaiaad6gaae qaniabggHiLdGccqGH+aGpcaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabodacaqGPa aaaa@7192@ where I0 > =I0 > ex Io ph I0 > ex and I0 > ph are the ground states of the exciton and phonon, respectively, a q ( a q + ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGXbaabeaakiaacIcacaWGHbWaa0baaSqaaiaadghaaeaa cqGHRaWkaaGccaGGPaaaaa@3C55@ is annihilation (creation) operator of the phonon with wave vector q, ϕ n (t) and  β n (t)=<Φ| u n |Φ> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaGaaeiiaiaabgga caqGUbGaaeizaiaabccacqaHYoGydaWgaaWcbaGaaeOBaaqabaGcca GGOaGaamiDaiaacMcacqGH9aqpcqGH8aapcqqHMoGrcaGG8bGaamyD amaaBaaaleaacaWGUbaabeaakiaacYhacqqHMoGrcqGH+aGpaaa@4E7D@ and π n (t)=<|Φ| P n |Φ>  MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0Jaeyip aWJaaiiFaiabfA6agjaacYhacaWGqbWaaSbaaSqaaiaad6gaaeqaaO GaaiiFaiabfA6agjabg6da+iaabccaaaa@46D4@ α nq (t)=< D 1 (t)| a q | D 1 (t)> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaad6gacaWGXbaabeaakiaacIcacaWG0bGaaiykaiabg2da 9iabgYda8iaadseadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDai aacMcacaGG8bGaamyyamaaBaaaleaacaWGXbaabeaakiaacYhacaWG ebWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadshacaGGPaGaeyOpa4 daaa@4B41@ are some undetermined functions of time. Obviously, ϕ D (t)>= n ϕ n (t) B n + 10 > ex MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadseaaeqaaOGaaiikaiaadshacaGGPaGaeyOpa4Jaeyyp a0ZaaabuaeaacqaHvpGzdaWgaaWcbaGaamOBaaqabaGccaGGOaGaam iDaiaacMcacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaaI XaGaaGimaaWcbaGaamOBaaqab0GaeyyeIuoakiabg6da+maaBaaale aacaWGLbGaamiEaaqabaaaaa@4CE2@ in Eq.(2) is an eigenstate of the number operator N ^ = n B n + B n , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja Gaeyypa0ZaaabuaeaacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWk aaGccaWGcbWaaSbaaSqaaiaad6gaaeqaaaqaaiaad6gaaeqaniabgg HiLdGccaGGSaaaaa@405C@ corresponding to a single excitation, i.e., N ^ | ϕ D (t)>=| ϕ D (t)> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja GaaiiFaiabew9aMnaaBaaaleaacaWGebaabeaakiaacIcacaWG0bGa aiykaiabg6da+iabg2da9iaacYhacqaHvpGzdaWgaaWcbaGaamiraa qabaGccaGGOaGaamiDaiaacMcacqGH+aGpaaa@4620@ .

The Davydov soliton obtained from Eqs. (1)-(2) in the semi classical limit and using the continuum approximation has the from ϕ D (x,t)= ( μ D 2 ) 1/2 sech[ μ D r 0 (x x 0 vt) ]exp{ i[ v 2J r 0 2 (x x 0 ) E ν t/) ] }     (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadseaaeqaaOGaaiikaiaadIhacaGGSaGaamiDaiaacMca cqGH9aqpdaqadaqaamaalaaabaGaeqiVd02aaSbaaSqaaiaadseaae qaaaGcbaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigda caGGVaGaaGOmaaaakiaabohacaqGLbGaae4yaiaabIgadaWadaqaam aalaaabaGaeqiVd02aaSbaaSqaaiaadseaaeqaaaGcbaGaamOCamaa BaaaleaacaaIWaaabeaaaaGccaGGOaGaamiEaiabgkHiTiaadIhada WgaaWcbaGaaGimaaqabaGccqGHsislcaWG2bGaamiDaiaacMcaaiaa wUfacaGLDbaaciGGLbGaaiiEaiaacchadaGadaqaaiaadMgadaWada qaamaalaaabaGaeS4dHGMaamODaaqaaiaaikdacaWGkbGaamOCamaa DaaaleaacaaIWaaabaGaaGOmaaaaaaGccaGGOaGaamiEaiabgkHiTi aadIhadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyOeI0Iaamyramaa BaaaleaacqaH9oGBaeqaaOGaamiDaiaac+cacqWIpecAcaGGPaaaca GLBbGaayzxaaaacaGL7bGaayzFaaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeinaiaabMcaaaa@776D@ Corresponding to an excitation localized over a scale r/ Dμ , where μ D = χ 1 2 (1 s 2 )wJ ,    G D =4J μ D ,     s 2 = v 2 v 0 2 ,     v 0 = r 0 (w/M) 1/2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadseaaeqaaOGaeyypa0ZaaSaaaeaacqaHhpWydaqhaaWc baGaaGymaaqaaiaaikdaaaaakeaacaGGOaGaaGymaiabgkHiTiaado hadaahaaWcbeqaaiaaikdaaaGccaGGPaGaam4DaiaadQeaaaGaaiil aiaabccacaqGGaGaaeiiaiaabEeadaWgaaWcbaGaaeiraaqabaGccq GH9aqpcaaI0aGaamOsaiabeY7aTnaaBaaaleaacaWGebaabeaakiaa cYcacaqGGaGaaeiiaiaabccacaqGGaGaae4CamaaCaaaleqabaGaae Omaaaakiabg2da9maalaaabaGaamODamaaCaaaleqabaGaaGOmaaaa aOqaaiaadAhadaqhaaWcbaGaaGimaaqaaiaaikdaaaaaaOGaaiilai aabccacaqGGaGaaeiiaiaabccacaqG2bWaaSbaaSqaaiaabcdaaeqa aOGaeyypa0JaamOCamaaBaaaleaacaqGWaaabeaakiaabIcacaqG3b Gaae4laiaab2eacaqGPaWaaWbaaSqabeaacaqGXaGaae4laiaabkda aaaaaa@666D@ is the sound speed in the protein molecular chains, v is the velocity of the soliton, r0 is the lattice constant. Evidently, the soliton contains only one exciton, i.e., ϕ D (t)| N ^ | ϕ D (t)>=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadseaaeqaaOGaaiikaiaadshacaGGPaWaaqWaaeaaceWG obGbaKaaaiaawEa7caGLiWoacqaHvpGzdaWgaaWcbaGaamiraaqaba GccaGGOaGaamiDaiaacMcacqGH+aGpcqGH9aqpcaaIXaaaaa@46F5@ This shows that the Davydov soliton is formed through self-trapping of one exciton with binding energy EBD, E BD = χ 1 4 3J w 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiraaqabaGccqGH9aqpdaWcaaqaaiabgkHiTiab eE8aJnaaDaaaleaacaaIXaaabaGaaGinaaaaaOqaaiaaiodacaWGkb Gaam4DamaaCaaaleqabaGaaGOmaaaaaaaaaa@4160@

Davydov’s idea yields a compelling picture for the mechanism of bioenergy transport in protein molecules and consequently has been the subject of a large number of works [3-28]. A lot of issues related to the Davydov model, including the foundation and accuracy of the theory, the quantum and classical properties and the thermal stability and lifetimes of the Davydov soliton have extensively been studied by many scientists [7-26]. However, considerable controversy has arisen concerning whether the Davydov soliton is sufficiently stable in the region of biological temperature to provide a viable explanation for bio-energy transport. It is out of question that the quantum fluctuations and thermal perturbations are expected to cause the Davydov soliton to decay into a delocalized state. Some numerical simulations indicated that the Davydov soliton is not stable at the biological temperature 300K [7-11, 24-26]. Other simulations showed that the Davydov soliton is stable at 300 K [10-24], but they were based on classical equations of motion which are likely to yield unreliable estimates for the stability of the soliton [3]. The simulations based on the ID2 state in Eq.(2) generally show that the stability of the soliton decreases with increasing temperatures and that the soliton is not sufficiently stable in the region of biological temperature. Since the dynamical equations used in the simulations are not equivalent to the equation, the stability of the soliton obtained by these numerical simulations is unavailable or unreliable. The simulation[9] based on the ID1>state in Eq. (3) with the thermal treatment of Davydov[3], where the equations of motion are derived from a thermally averaged Hamiltonian, yields the confusing result that the stability of the soliton is enhanced with increasing temperature, predicting that ID1>- type soliton is stable in the region of biological temperature. Evidently, the conclusion is doubtful because the Davydov procedure, in which an equation of motion for an average dynamical state from an average Hamiltonian, corresponding to the Hamiltonian averaged over a thermal distribution of phonons, is inconsistent with standard concepts of quantum-statistical mechanics in which a density matrix must be used to describe the system. Therefore, any exact fully quantum- mechanical treatment for the numerical simulation of the Davydov soliton does not exist. However, for the thermal equilibrium properties of the Davydov soliton, there is a quantum Monte Carlo simulation [13]. In the simulation, correlation characteristic of solitonlike quasiparticles occur only at low temperatures, about T< 10k, for widely accepted parameter values. This is consistent at a qualitative level with the result of Cottingham et al. [15,21]. The latter is a straightforward quantum-mechanical perturbation calculation. The lifetime of the Davydov soliton obtained by using this method is too small (about 10-12 - 10-13Sec) to be useful in biological processes. This indicates clearly that the Davydov solution is not a true wave function of the systems. A through study in terms of parameter values, different types of disorder, different thermalization schemes, different wave functions, and different associated dynamics leads to a very complicated picture for the Davydov model [10-12]. These results do not completely rule out the Davydov theory, however they do not eliminate the possibility of another wave function and a more sophisticated Hamiltonian of the system having a soliton with longer lifetimes and good thermal stability.

Indeed, the question of the lifetime of the soliton in protein molecules is twofold. In Langevin dynamics, the problem consists of uncontrolled effects arising from the semiclassical approximation. In quantum treatments, the problem has been the lack of an exact wave function for the soliton. The exact wave function of the fully quantum Davydov model has not been known up to now. Different wave functions have been used to describe the states of the fully quantum-mechanical systems [4,5]. Although some of these wave functions lead to exact quantum states and exact quantum dynamics in the J=0 state, they also share a problem with the original Davydov wave function, namely that the degree of approximation included when J0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabgc Mi5kaaicdaaaa@3945@ is not known. Therefore, it is necessary to reform Davydov’s wave function. Scientists had though that the soliton with a multiquantum n2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOBaiabgw MiZkaabkdaaaa@3961@ , for example, the coherent state of Brown et al. [4], the multiquantum state of Kerr et al. [12] and Schweitzer et.al. [15,21], the two-quantum state of Cruzeiro-Hansson [18] and Forner [22], and so on, would be thermally stable in the region of biological temperature and could provide a realistic mechanism for bioenergy transport in protein molecules. However, the assumption of the standard coherent state is unsuitable or impossible for biological protein molecules because there are innumerable particles in this state and one could not retain conservation of the number of particles of the system. The assumption of a multiquantum state (n>2) along with a coherent state is also inconsistent with the fact that the bioenergy released in ATP hydrolysis can excite only two quanta of amide-I vibration. On the other hand, the numerical result shows that the soliton of two-quantum state is more stable than that with a one-quantum state.

Cruzeiro-Hansson [18] had thought that Forner’s twoquantum state in the semiclassical case was not exact. Therefore, he constructed again a so-called exactly two-quantum state for the semiclassical Davydov system as follows [18]: Iϕ( t )>= n,m=1 N φ nm ( { u 1 },{ P 1 },t ) B n + B m + | 0 > ex ,      (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeysaiabew 9aMnaabmaabaGaaeiDaaGaayjkaiaawMcaaiabg6da+iabg2da9maa qahabaGaeqOXdO2aaSbaaSqaaiaab6gacaqGTbaabeaaaeaacaqGUb Gaaiilaiaab2gacqGH9aqpcaqGXaaabaGaaeOtaaqdcqGHris5aOWa aeWaaeaacaGG7bGaaeyDamaaBaaaleaacaqGXaaabeaakiaac2haca GGSaGaai4EaiaabcfadaWgaaWcbaGaaeymaaqabaGccaGG9bGaaiil aiaabshaaiaawIcacaGLPaaacaqGcbWaa0baaSqaaiaab6gaaeaacq GHRaWkaaGccaqGcbWaa0baaSqaaiaab2gaaeaacaqGRaaaaOWaaqqa aeaacaqGWaGaeyOpa4ZaaSbaaSqaaiaabwgacaqG4baabeaaaOGaay 5bSdGaaiilaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bIcacaqG1aGaaeykaaaa@651F@ where B n ( B n + ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWGUbaabeaakiaacIcacaWGcbWaa0baaSqaaiaad6gaaeaa cqGHRaWkaaGccaGGPaaaaa@3C11@ is the annihilation (creation) operator for an amide-I vibration quantum (exciton), u 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaaIXaaabeaaaaa@37D6@ is the displacement of the lattice molecules, P 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaaIXaaabeaaaaa@37B1@ is its conjugate momentum, and |0 ex MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaaGimaaGaay5bSlaawQYiamaaBaaaleaacaWGLbGaamiEaaqabaaa aa@3B58@ is the ground state of the exciton. He calculate the average probability distribution of the exciton per site, and average displacement difference per site, and the thermodynamics average of the variable, P= B 1 + B 1 B 2 + B 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabg2 da9iaadkeadaqhaaWcbaGaaGymaaqaaiabgUcaRaaakiaadkeadaWg aaWcbaGaaGymaaqabaGccqGHsislcaWGcbWaa0baaSqaaiaaikdaae aacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaaikdaaeqaaaaa@415B@ as a measure of localization of the exciton, versus quantity ν=JW/ χ 1 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaamOsaiaadEfacaGGVaGaeq4Xdm2aa0baaSqaaiaaigdaaeaa caaIYaaaaaaa@3E6C@ and lnβ(β=1/ K B T) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaad6 gacqaHYoGycaGGOaGaeqOSdiMaeyypa0JaaGymaiaac+cacaWGlbWa aSbaaSqaaiaadkeaaeqaaOGaamivaiaacMcaaaa@418E@ in the so-called twoquantum state. Eq.(5), where χ 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaaaa@3893@ is a nonlinear coupling parameter related to the interaction of the exciton-phonon in the Davydov model. Their energies and stability are compared with those of the one-quantum state. From the results of above thermal averages, he drew the conclusion that the wave function with a two-quantum state can lead to more stable soliton solutions than that with a one-quantum state, and that the usual Langevin dynamics ,whereby the thermal lifetime of the Davydov soliton is estimated, must be viewed as underestimating the soliton lifetime.

However, by checking carefully Eq.(5), we can find that the Cruzeiro-Hansson wave function[18,24-26] does not represent exactly the two-quantum state. To find out how many quanta the state Eq.(1), indeed contains, the expectation value of the exciton number operator has to be computed. N= n B n + B n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maaqababaGaamOqaaWcbaGaamOBaaqab0GaeyyeIuoakmaaDaaa leaacaWGUbaabaGaey4kaScaaOGaamOqamaaBaaaleaacaWGUbaabe aaaaa@3F67@ , in this state Eq.(5), and sum over the sites, i.e., the exciton numbers N are N=<ϕ| n B n + B n | ϕ> = ijlmn φ im φ jl ex<0| B i B m B n + B n B j + B l + |0 > ex       (6)   = nj ( φ nj φ jn + φ jn φ jn ) + nl ( φ nl φ nl + φ ln φ nl ) =4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGceaqabeaacaWGob Gaeyypa0JaeyipaWJaeqy1dy2aaqqaaeaadaaeqbqaaiaadkeadaqh aaWcbaGaamOBaaqaaiabgUcaRaaaaeaacaWGUbaabeqdcqGHris5aa GccaGLhWoacaWGcbWaaSbaaSqaaiaad6gaaeqaaOWaaqqaaeaacqaH vpGzcqGH+aGpaiaawEa7aiabg2da9maaqafabaGaeqOXdO2aa0baaS qaaiaadMgacaWGTbaabaGaey4fIOcaaaqaaiaadMgacaWGQbGaamiB aiaad2gacaWGUbaabeqdcqGHris5aOGaeqOXdO2aaSbaaSqaaiaadQ gacaWGSbaabeaakiaadwgacaWG4bGaeyipaWJaaGimamaaeeaabaGa amOqamaaBaaaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyBaa qabaGccaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWGcbWa aSbaaSqaaiaad6gaaeqaaOGaamOqamaaDaaaleaacaWGQbaabaGaey 4kaScaaOGaamOqamaaDaaaleaacaWGSbaabaGaey4kaScaaaGccaGL hWoadaabbaqaaiaaicdaaiaawEa7aiabg6da+maaBaaaleaacaWGLb GaamiEaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caGGOaGaaGOnaiaacMcaaeaacaqGGaGaaeiiaiabg2da9maaqafaba WaaeWaaeaacqaHgpGAdaqhaaWcbaGaamOBaiaadQgaaeaacqGHxiIk aaGccqaHgpGAdaWgaaWcbaGaamOAaiaad6gaaeqaaOGaey4kaSIaeq OXdO2aa0baaSqaaiaadQgacaWGUbaabaGaey4fIOcaaOGaeqOXdO2a aSbaaSqaaiaadQgacaWGUbaabeaaaOGaayjkaiaawMcaaaWcbaGaam OBaiaadQgaaeqaniabggHiLdGccqGHRaWkdaaeqbqaamaabmaabaGa eqOXdO2aa0baaSqaaiaad6gacaWGSbaabaGaey4fIOcaaOGaeqOXdO 2aaSbaaSqaaiaad6gacaWGSbaabeaakiabgUcaRiabeA8aQnaaDaaa leaaciGGSbGaaiOBaaqaaiabgEHiQaaakiabeA8aQnaaBaaaleaaca WGUbGaamiBaaqabaaakiaawIcacaGLPaaaaSqaaiaad6gacaWGSbaa beqdcqGHris5aOGaeyypa0JaaGinaaaaaa@ABFB@ where we use the relations [ B n . B j + ]= σ nj , nl | φ nl | 2 =1       (7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadk eadaWgaaWcbaGaamOBaaqabaGccaGGUaGaamOqamaaDaaaleaacaWG QbaabaGaey4kaScaaOGaaiyxaiabg2da9iabeo8aZnaaBaaaleaaca WGUbGaamOAaaqabaGccaGGSaWaaabeaeaadaabdaqaaiabeA8aQnaa BaaaleaacaWGUbGaamiBaaqabaaakiaawEa7caGLiWoadaahaaWcbe qaaiaaikdaaaaabaGaamOBaiaadYgaaeqaniabggHiLdGccqGH9aqp caaIXaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabIcacaqG3aGaaeykaaaa@56A6@ ex 0| B n + |0 ex = ex 0| B n + B n |0 ex = ex 0| B n + B n B l |0 ex =....=0         (8) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaWaaSbaaSqaai aadwgacaWG4baabeaakmaaEeaabaGaaGimaaqabiaawMYicaGLhWoa caWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGcdaGhcaqabeaaca aIWaaacaGLhWUaayPkJaWaaSbaaSqaaiaadwgacaWG4baabeaakiab g2da9maaBaaaleaacaWGLbGaamiEaaqabaGcdaGhbaqaaiaaicdaae qacaGLPmIaay5bSdGaamOqamaaDaaaleaacaWGUbaabaGaey4kaSca aOGaamOqamaaBaaaleaacaWGUbaabeaakmaaEiaabeqaaiaaicdaai aawEa7caGLQmcadaWgaaWcbaGaamyzaiaadIhaaeqaaOGaeyypa0Za aSbaaSqaaiaadwgacaWG4baabeaakmaaEeaabaGaaGimaaqabiaawM YicaGLhWoacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWG cbWaaSbaaSqaaiaad6gaaeqaaOGaamOqamaaBaaaleaacaWGSbaabe aakmaaEiaabeqaaiaaicdaaiaawEa7caGLQmcadaWgaaWcbaGaamyz aiaadIhaaeqaaOGaeyypa0JaaiOlaiaac6cacaGGUaGaaiOlaiabg2 da9iaaicdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaGGOaGaaGioaiaacMcaaaa@7435@ Therefore, the state Eq.(5), as it is put forward in Ref. [10], deals with four excitons (quanta), instead of two excitons, in contradiction to the author’s statements. Obviously, it is impossible to create the four excitons by the energy released in the ATP hydrolysis (about 0.43 eV). Thus the author’s wave function is still not relevant to protein molecules, and his discussion and conclusion are all unreliable and implausible in that paper [10].

It is believed that the physical significance of the wave function, Eq. (5), is also unclear, or at least is very difficult to understand. As far as the physical meaning of Eq.(5) is concerned, it represents only a combinational state of single-particle excitation with two quanta created at sites n and m;[18,26] is the probability amplitude of particles occurring at the sites n and m simultaneously. In general, n=≠m and in accordance with the author’s idea. In such a case it is very difficult to imagine the form of the soliton by the mechanism of self- trapping of the two quanta under the action of the nonlinear exciton-phonon interaction, especially when the difference between n and m is very large. Hansson has also not explained the physical and biological reasons and the meaning for the proposed trial state. Therefore, we think that the Cruzeiro-Hansson representation is still not an exact wave function suitable for protein molecules. Thus, the wave function of the systems is still an open problem today.

On the basis of the work of Cruzeio-Hansson [12,13,18,22], Schweitzer [21] and Pang [24-26] proposed a new model of the bioenergy transport in the protein molecules, in which both the Hamiltonian and the wave function of the Dovydov model [24] have been improved. A new coupling interaction between the acoustic and amide-I vibrational modes was added to the original Davydov’s Hamiltonian which takes into account relative displacement of the neighbouring amino acids resulting from dipole-dipole interaction of the neighbouring amide-1 vibrational quanta. Davydov’s wave function has been also replaced with a quasi-coherent two-quanta state to exhibit the coherent behaviors of collective excitations of the excitons and phonons [25-26] which are a feature of the energy released in ATP hydrolysis in the systems. The equation of motion and the properties of the new soliton in the new model are different from those in the Davydov model and as a result the soliton lifetime and stability are greatly enhanced. It is suggested that this model can resolve the controversy on the thermal stability and lifetime of the soliton excited in the protein molecules. The quantum properties of the new soliton will be studied here, but here attention is paid also to the problem of its lifetime and thermal stability at biological temperature 300 K and the lifetime of the new soliton at 300K is calculated in detail by using the generally accepted values of the parameters appropriate to -helical protein molecules in terms of the quantum perturbation theory developed by Cottingham et al. [15], which can take simultaneously into account the quantum and thermal effects. It can be seen that the lifetime of the new soliton at 300 K is long enough to provide a viable explanation of the bio-energy transport in the proteins. The plan of this paper is as follows. In Section 2, the new model, including the extended Hamiltonian and the wave function, is presented. The equations of motion and the new soliton solution in this model are given in Section 3. In Section 4, the properties and thermal stability of the new soliton are discussed, and the possibility of the soliton being a suitable candidate for the mechanism of bioenergy transport in protein molecules is predicted on the basis of results obtained in this paper. In Section 5, the properties of the new soliton are described and its lifetime is calculated by using quantummechanical perturbution method. The detailed discussion of the properties and changes of the lifetimes of the soliton for a large range of parameter values is presented. The conclusions of this investigation are also given in this section.
III. Eatablishment of new theory of Bio-energy transport in the Protein molecules
Results obtained by many scientists over the years indicate that the Davydov model, whether it is the wave function or the Hamiltonian, is indeed too simple, i.e.., it does not denote the elementary properties of the collective excitations occurring in protein molecules, and many improvements of it have been unsuccessful, as mentioned above. What is the source of this problem? It is well known that the Davydov theory on bioenergy transport was introduced into protein molecules from an excitonsoliton model in generally one-dimensional molecular chains [24]. Although the molecular structure of the alpha-helix protein is analogous to some molecular crystals, for example acetanilide (ACN) (in fact, both are polypeptides; the alpha-helix protein molecule is the structure of three peptide channels, ACN is the structure of two peptide channels. If comparing the structure of alpha helix protein with ACN, we find that the hydrogen-boned peptide channels with the atomic structure along the longitudinal direction are the same except for the side group), a lot of properties and functions of the protein molecules are completely different from that of the latter. The protein molecules are both a kinds of soft condensed matter and bio-self-organization with action functions, for instance, self-assembling and self-renovating. The physical concepts of coherence, order, collective effects, and mutual correlation are very important in bio-self- organization, including the protein molecules, when compared with generally molecular systems [25.26]. Therefore, it is worth studying how we can physically describe these properties. It is noted that Davydov operation is not strictly correct. Therefore, it is believed that a basic reason for the failure of the Davydov model is just that it ignores completely the above important properties of the protein molecules.

Let us consider the Davydov model with the present viewpoint. First, as far as the Davydov wave functions, both | D 1 >and| D 2 > MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqqaaeaaca WGebWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWoacqGH+aGpcaWGHbGa amOBaiaadsgadaabbaqaaiaadseadaWgaaWcbaGaaGOmaaqabaGccq GH+aGpaiaawEa7aaaa@4164@ are concerned [3], they are not true solutions of the protein molecules. On the one hand, there is obviously asymmetry in the Davydov wave function since the phononic parts is a coherent state, while the excitonic part is only an excitation state of a single particle. It is not reasonable that the same nonlinear interaction generated by the coupling between the excitons and phonons produces different states for the phonon and exciton. Thus, Davydov’s wave function should be modified [24-26], i.e., the excitonic part in it should also be coherent or quasicoherent to represent the coherent feature of collective excitation in protein molecules. However, the standard coherent [4] and large-n excitation states [12,22] are not appropriate for the protein molecules due to the reasons mentioned above. Similarly, Forner’s and Cruzeiro-Hansson’s two-quantum states do not fulfill the above request. In view of the above discussion, we proposed the following wave function of the protein molecular systems: Φ(t)= |φ P ( t )>| β( t ) >= 1 λ [ I+ n φ n ( t ) B n + + 1 2! ( n φ n ( t ) B n + ) 2 ]| 0 > ex ×      (9) exp( i n [ β n (t) P n π n (t) u n ] ) |0 ph MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHMo GrcaGGOaGaamiDaiaacMcacqGH9aqpdaabbaqaaiabeA8aQbGaay5b SdWaaSbaaSqaaiaadcfaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaeyOpa4ZaaqqaaeaacqaHYoGydaqadaqaaiaadshaaiaawIca caGLPaaaaiaawEa7aiabg6da+iabg2da9maalaaabaGaaGymaaqaai abeU7aSbaadaWadaqaaiaadMeacqGHRaWkdaaeqbqaaiabeA8aQnaa BaaaleaacaWGUbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaai aadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiabgUcaRmaalaaa baGaaGymaaqaaiaaikdacaGGHaaaaaWcbaGaamOBaaqab0GaeyyeIu oakmaabmaabaWaaabuaeaacqaHgpGAdaWgaaWcbaGaamOBaaqabaGc daqadaqaaiaadshaaiaawIcacaGLPaaacaWGcbWaa0baaSqaaiaad6 gaaeaacqGHRaWkaaaabaGaamOBaaqab0GaeyyeIuoaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5waiaaw2faamaaeeaaba GaaGimaiabg6da+maaBaaaleaacaWGLbGaamiEaaqabaaakiaawEa7 aiabgEna0kaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabI cacaqG5aGaaeykaaqaaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOe I0YaaSaaaeaacaWGPbaabaGaeS4dHGgaamaaqafabaGaai4waiabek 7aInaaBaaaleaacaWGUbaabeaakiaacIcacaWG0bGaaiykaiaadcfa daWgaaWcbaGaamOBaaqabaGccqGHsislcqaHapaCdaWgaaWcbaGaam OBaaqabaGccaGGOaGaamiDaiaacMcacaWG1bWaaSbaaSqaaiaad6ga aeqaaOGaaiyxaaWcbaGaamOBaaqab0GaeyyeIuoaaOGaayjkaiaawM caamaaEiaabeqaaiaaicdaaiaawEa7caGLQmcadaWgaaWcbaGaamiC aiaadIgaaeqaaaaaaa@9B4C@ where B n + MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaDa aaleaacaWGUbaabaGaey4kaScaaaaa@38BE@ and B n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWGUbaabeaaaaa@37DB@ are boson creation and annihilation operators for the exciton, | 0 > ex and| 0 > ph MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqqaaeaaca aIWaGaeyOpa4ZaaSbaaSqaaiaadwgacaWG4baabeaaaOGaay5bSdGa amyyaiaad6gacaWGKbWaaqqaaeaacaaIWaGaeyOpa4ZaaSbaaSqaai aadchacaWGObaabeaaaOGaay5bSdaaaa@4398@ are the ground states of the exciton and phonon, respectively u n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGUbaabeaaaaa@380E@ and P n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGUbaabeaaaaa@37E9@ are the displacement and momentum operators of the amino acid residue at site n respectively. The ϕ n ( t ). β n ( t )=<Φ( t )| u n | Φ( t )>and π n ( t )=<Φ( t )| P n |Φ( T )> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGa aiOlaiabek7aInaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiDaa GaayjkaiaawMcaaiabg2da9iabgYda8iabfA6agnaabmaabaGaamiD aaGaayjkaiaawMcaamaaeeaabaGaamyDamaaBaaaleaacaWGUbaabe aaaOGaay5bSdWaaqqaaeaacqqHMoGrdaqadaqaaiaadshaaiaawIca caGLPaaacqGH+aGpcaWGHbGaamOBaiaadsgacqaHapaCdaWgaaWcba GaamOBaaqabaaakiaawEa7amaabmaabaGaamiDaaGaayjkaiaawMca aiabg2da9iabgYda8iabfA6agnaabmaabaGaamiDaaGaayjkaiaawM caamaaemaabaGaamiuamaaBaaaleaacaWGUbaabeaaaOGaay5bSlaa wIa7aiabfA6agnaabmaabaGaamivaaGaayjkaiaawMcaaiabg6da+a aa@69EF@ are there sets of unknown functions, λ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37A9@ is a normalization constant. It is assumed hereafter that λ=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey ypa0JaaGymaaaa@396A@ for convenience of calculation, except when explicitly mentioned.

A second problem arises for the Davydov Hamiltonian [24-26,28]. The Davydov Hamiltonian takes into account the resonant or dipole-dipole interaction of the neighboring amide-I vibrational quanta in neighboring amino acid residues with an electrical moment of about 3.5D, but why do we not consider the changes of relative displacement of the neighboring amino acid residues arising from this interaction ? It is reasonable to add the new interaction term χ 2 ( u n+1 u n )( B n+1 + B n + B m + B n+1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaikdaaeqaaOWaaeWaaeaacaWG1bWaaSbaaSqaaiaad6ga cqGHRaWkcaaIXaaabeaakiabgkHiTiaadwhadaWgaaWcbaGaamOBaa qabaaakiaawIcacaGLPaaadaqadaqaaiaadkeadaqhaaWcbaGaamOB aiabgUcaRiaaigdaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaad6 gaaeqaaOGaey4kaSIaamOqamaaDaaaleaacaWGTbaabaGaey4kaSca aOGaamOqamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaakiaawI cacaGLPaaaaaa@5021@ into the Davydov’s Hamiltonian to represent correlations of the collective excitations and collective motions in the protein molecules, as mentioned above [24-26]. Although the dipole- dipole interaction is small as compared with the energy of the amide-I vibrational quantum, the change of relative displacement of neighboring peptide groups resulting from this interaction cannot be ignored due to the sensitive dependence of dipole-dipole interaction on the distance between amino acids in the protein molecules, which is a kind of soft condensed matter and bio-self-organization. Thus, the Davydov Hamiltonian is replaced by H= H ex + H ph + H int = n [ ε 0 B n + B n J( B n + B n+1 + B n B n+1 + ) ] + n [ P n 2 2M + 1 2 w ( u n u n1 ) 2 ] + n [ χ 1 ( u n+1 u n1 ) ] B n + B n + χ 2 ( u n+1 u n )×( B n+1 + B n + B n + B n+1 )                        (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGib Gaeyypa0JaamisamaaBaaaleaacaWGLbGaamiEaaqabaGccqGHRaWk caWGibWaaSbaaSqaaiaadchacaWGObaabeaakiabgUcaRiaadIeada WgaaWcbaGaciyAaiaac6gacaGG0baabeaakiabg2da9maaqafabaWa amWaaeaacqaH1oqzdaWgaaWcbaGaaGimaaqabaGccaWGcbWaa0baaS qaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaad6gaaeqa aOGaeyOeI0IaamOsamaabmaabaGaamOqamaaDaaaleaacaWGUbaaba Gaey4kaScaaOGaamOqamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqa baGccqGHRaWkcaWGcbWaa0baaSqaaiaad6gaaeaaaaGccaWGcbWaa0 baaSqaaiaad6gacqGHRaWkcaaIXaaabaGaey4kaScaaaGccaGLOaGa ayzkaaaacaGLBbGaayzxaaaaleaacaWGUbaabeqdcqGHris5aOGaey 4kaSYaaabeaeaacaGGBbWaaSaaaeaacaWGqbWaa0baaSqaaiaad6ga aeaacaaIYaaaaaGcbaGaaGOmaiaad2eaaaGaey4kaSIaaGjcVpaala aabaGaaGymaaqaaiaaikdaaaaaleaacaWGUbaabeqdcqGHris5aOGa am4DaiaacIcacaWG1bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iaam yDamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaOGaaiyxaaqaaiabgUcaRmaaqafabaWaamWaae aacqaHhpWydaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiaadwhadaWg aaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyOeI0IaamyDamaaBa aaleaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaiaa wUfacaGLDbaaaSqaaiaad6gaaeqaniabggHiLdGccaWGcbWaa0baaS qaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaad6gaaeqa aOGaey4kaSIaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaaca WG1bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTiaa dwhadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaacqGHxdaTda qadaqaaiaadkeadaqhaaWcbaGaamOBaiabgUcaRiaaigdaaeaacqGH RaWkaaGccaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaamOqam aaDaaaleaacaWGUbaabaGaey4kaScaaOGaamOqamaaBaaaleaacaWG UbGaey4kaSIaaGymaaqabaaakiaawIcacaGLPaaacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGWaGaaeykaaaaaa@CE1E@ Where ε 0 =0.205ev MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIYaGaaGim aiaaiwdacaqGLbGaaeODaaaa@3F14@ is the energy of the exciton (the C=0 strechiong mode). The present nonlinear coupling constants are χ 1 and χ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaOGaamyyaiaad6gacaWGKbGaeq4Xdm2aaSba aSqaaiaaikdaaeqaaaaa@3DFE@ They represent the modulations of the on-site energy and resonant (or dipole-dipole) interaction energy of excitons caused by the molecules displacements, respectively .M is the mass of a amino acid molcule and w is the elasticity constant of the protein molecular chains. J is the dipole-dipole interaction energy between neighboring sites. The physical meaning of the other quantities in Eq.(6)are the same as those in the above explanations.

The Hamiltonian and wave function shown in Eqs. (9)- (10) are different from Davydov’s. We add a new interaction term n χ 2 ( u n+1 u n )( B n+1 + B n + B n + B n+1 ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aHhpWydaWgaaWcbaGaaGOmaaqabaaabaGaamOBaaqab0GaeyyeIuoa kmaabmaabaGaamyDamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqaba GccqGHsislcaWG1bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzk aaWaaeWaaeaacaWGcbWaa0baaSqaaiaad6gacqGHRaWkcaaIXaaaba Gaey4kaScaaOGaamOqamaaBaaaleaacaWGUbaabeaakiabgUcaRiaa dkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiaadkeadaWgaaWcba GaamOBaiabgUcaRiaaigdaaeqaaaGccaGLOaGaayzkaaGaaiilaaaa @539D@ into the original Davydov Hamiltonian. Thus the Hamiltonian now has better correspondence between the interactions and can also represent the features of mutual correlations of the collective excitations and of collective motions in the protein molecules. We should point out here that the different coupling between the relevant modes was also considered by Pang [24-26] and others [27-28] in the Hamiltonian of the vibron-soliton model for one-dimensional oscillator-lattice and protein systems, respectively, but the wave functions of the systems they used are different from Eqs. (9)- (10).

Evidently , the present wave function of the exciton in Eq.(9) is not an excitation state of a single particle, but rather a coherent state, more precisely, a quasicoherent state, because it retain only for three terms of the expansion of a standard coherent state, which can be viewed as an effective truncation of a standard coherent state. When φ n ( t ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aaS baaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaa aa@3B5D@ is small, i.e., | φ n ( t ) |<<1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq aHgpGAdaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadshaaiaawIca caGLPaaaaiaawEa7caGLiWoacqGH8aapcqGH8aapcaaIXaaaaa@4142@ Pang represented the wave function of the excitons, | φ P ( t )> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqqaaeaacq aHgpGAdaWgaaWcbaGaamiuaaqabaGcdaqadaqaaiaadshaaiaawIca caGLPaaacqGH+aGpaiaawEa7aaaa@3DDB@ , in Eq.(9) as | φ P ( t ) >= 1 λ [ 1+ n φ n ( t ) B n + + 1 2! ( n φ n ( t ) B n + ) 2 ]| 0 > ex ~ 1 λ exp[ 1 2 n | φ n (t) | 2 ]× exp{ n φ n ( t ) B n + }| 0 > ex = 1 λ exp{ n [ φ n ( t ) B n + φ n ( t ) B n ] }| 0 > ex                       (11) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaabba qaaiabeA8aQnaaBaaaleaacaWGqbaabeaakmaabmaabaGaamiDaaGa ayjkaiaawMcaaaGaay5bSdGaeyOpa4Jaeyypa0ZaaSaaaeaacaaIXa aabaGaeq4UdWgaamaadmaabaGaaGymaiabgUcaRmaaqafabaGaeqOX dO2aaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaamOqamaaDaaaleaacaWGUbaabaGaey4kaScaaOGaey4kaSYa aSaaaeaacaaIXaaabaGaaGOmaiaacgcaaaWaaeWaaeaadaaeqbqaai abeA8aQnaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiDaaGaayjk aiaawMcaaiaadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaaaeaaca WGUbaabeqdcqGHris5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI Yaaaaaqaaiaad6gaaeqaniabggHiLdaakiaawUfacaGLDbaadaabba qaaiaaicdacqGH+aGpdaWgaaWcbaGaamyzaiaadIhaaeqaaaGccaGL hWoacaGG+bWaaSaaaeaacaaIXaaabaGaeq4UdWgaaiGacwgacaGG4b GaaiiCaiaacUfacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaa qafabaWaaqWaaeaacqaHgpGAdaWgaaWcbaGaamOBaaqabaGccaGGOa GaamiDaiaacMcaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaa baGaamOBaaqab0GaeyyeIuoakiaac2facqGHxdaTaeaaciGGLbGaai iEaiaacchadaGadaqaamaaqafabaGaeqOXdO2aaSbaaSqaaiaad6ga aeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaamOqamaaDaaale aacaWGUbaabaGaey4kaScaaaqaaiaad6gaaeqaniabggHiLdaakiaa wUhacaGL9baadaabbaqaaiaaicdacqGH+aGpdaWgaaWcbaGaamyzai aadIhaaeqaaaGccaGLhWoacqGH9aqpdaWcaaqaaiaaigdaaeaacqaH 7oaBaaGaciyzaiaacIhacaGGWbWaaiWaaeaadaaeqbqaamaadmaaba GaeqOXdO2aaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaamOqamaaDaaaleaacaWGUbaabaGaey4kaScaaOGaey OeI0IaeqOXdO2aa0baaSqaaiaad6gaaeaacqGHxiIkaaGcdaqadaqa aiaadshaaiaawIcacaGLPaaacaWGcbWaaSbaaSqaaiaad6gaaeqaaa GccaGLBbGaayzxaaaaleaacaWGUbaabeqdcqGHris5aaGccaGL7bGa ayzFaaWaaqqaaeaacaaIWaGaeyOpa4ZaaSbaaSqaaiaadwgacaWG4b aabeaaaOGaay5bSdGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeymaiaabMca aaaa@D22E@ The last representation in Eq.(11) is a standard coherent state. Therefore, the state of exciton denoted by the new wave function | φ( t ) > MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqqaaeaacq aHgpGAdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawEa7aiabg6da +aaa@3CD0@ has a coherent feature, thus the wave function inEq.11) is normalized at λ=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey ypa0JaaGymaaaa@396A@ Since n | φ n ( t ) | 2 =1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaada abdaqaaiabeA8aQnaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaGaay5bSlaawIa7aaWcbaGaamOBaaqab0Gaey yeIuoakmaaCaaaleqabaGaaGOmaaaakiabg2da9iaaigdaaaa@4413@ required in thecalculation, then this condition of | φ n ( t ) |<<1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq aHgpGAdaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadshaaiaawIca caGLPaaaaiaawEa7caGLiWoacqGH8aapcqGH8aapcaaIXaaaaa@4142@ naturally satisfied for the roteins consisting of several hundreds of amino acids. Just so, the wave function denoted in Eq.(9) represents exactly the coherent features of collective excitations of both the excitons and phonons caused by the nonlinear exciton - phonon interaction, this indicates that the wave function in Eq.(9) is justified for the proteins. However, it is not an eigenstate of the number operator, N ^ = n B n + B n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja Gaeyypa0ZaaabeaeaacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWk aaGccaWGcbWaaSbaaSqaaiaad6gaaeqaaaqaaiaad6gaaeqaniabgg HiLdaaaa@3F62@ because of N ^ | φ P >= n B n + B n | φ P ={ n φ n ( t ) B n + + ( n φ n ( t ) B n + ) 2 }| 0 > ex =2| φ P ( 2+ n φ n ( t ) B n + )| 0 > ex (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja WaaqqaaeaacqaHgpGAdaWgaaWcbaGaamiuaaqabaaakiaawEa7aiab g6da+iabg2da9maaqafabaGaamOqamaaDaaaleaacaWGUbaabaGaey 4kaScaaaqaaiaad6gaaeqaniabggHiLdGccaWGcbWaaSbaaSqaaiaa d6gaaeqaaOWaa4HaaeqabaGaeqOXdO2aaSbaaSqaaiaadcfaaeqaaa GccaGLhWUaayPkJaGaeyypa0ZaaiWaaeaadaaeqbqaaiabeA8aQnaa BaaaleaacaWGUbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaai aadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaaaeaacaWGUbaabeqd cqGHris5aOGaey4kaSYaaeWaaeaadaaeqbqaaiabeA8aQnaaBaaale aacaWGUbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiaadkea daqhaaWcbaGaamOBaaqaaiabgUcaRaaaaeaacaWGUbaabeqdcqGHri s5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGL7bGa ayzFaaWaaqqaaeaacaaIWaGaeyOpa4ZaaSbaaSqaaiaadwgacaWG4b aabeaaaOGaay5bSdGaeyypa0JaaGOmamaaEiaabeqaaiabeA8aQnaa BaaaleaacaWGqbaabeaaaOGaay5bSlaawQYiaiabgkHiTmaabmaaba GaaGOmaiabgUcaRmaaqafabaGaeqOXdO2aaSbaaSqaaiaad6gaaeqa aOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaamOqamaaDaaaleaaca WGUbaabaGaey4kaScaaaqaaiaad6gaaeqaniabggHiLdaakiaawIca caGLPaaadaabbaqaaiaaicdacqGH+aGpdaWgaaWcbaGaamyzaiaadI haaeqaaaGccaGLhWoacaGGOaGaaGymaiaaikdacaGGPaaaaa@8BBF@ Therefore, the | φ P MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaeqOXdO2aaSbaaSqaaiaadcfaaeqaaaGccaGLhWUaayPkJaaaaa@3B53@ represents a coherent superposition of the excitonic state with two quanta and the ground state of the exciton, but it has a determinate numbers of quanta. From the expectation value of number operator N ^ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca WGobaacaGLcmaaaaa@378A@ we find that this state contains the number of exciton is N=< φ P | N ^ | φ P >= n φ P | B n + B n | φ P ={ n | φ n ( t ) | 2 +( n | φ n ( t ) | 2 )( m | φ m ( t ) | 2 ) } =( n | φ n ( t ) | 2 )( 1+ m | φ m ( t ) | 2 )=2                                                                    (13) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGob Gaeyypa0JaeyipaWJaeqOXdO2aaSbaaSqaaiaadcfaaeqaaOWaaqWa aeaaceWGobGbaKaaaiaawEa7caGLiWoacqaHgpGAdaWgaaWcbaGaam iuaaqabaGccqGH+aGpcqGH9aqpdaaeqbqaamaaEeaabaGaeqOXdO2a aSbaaSqaaiaadcfaaeqaaaGcbeGaayzkJiaawEa7aiaadkeadaqhaa WcbaGaamOBaaqaaiabgUcaRaaakiaadkeadaWgaaWcbaGaamOBaaqa baGcdaGhcaqabeaacqaHgpGAdaWgaaWcbaGaamiuaaqabaaakiaawE a7caGLQmcaaSqaaiaad6gaaeqaniabggHiLdGccqGH9aqpdaGadaqa amaaqafabaWaaqWaaeaacqaHgpGAdaWgaaWcbaGaamOBaaqabaGcda qadaqaaiaadshaaiaawIcacaGLPaaaaiaawEa7caGLiWoadaahaaWc beqaaiaaikdaaaGccqGHRaWkdaqadaqaamaaqafabaWaaqWaaeaacq aHgpGAdaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadshaaiaawIca caGLPaaaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaabaGaam OBaaqab0GaeyyeIuoaaOGaayjkaiaawMcaamaabmaabaWaaabuaeaa daabdaqaaiabeA8aQnaaBaaaleaacaWGTbaabeaakmaabmaabaGaam iDaaGaayjkaiaawMcaaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOm aaaaaeaacaWGTbaabeqdcqGHris5aaGccaGLOaGaayzkaaaaleaaca WGUbaabeqdcqGHris5aaGccaGL7bGaayzFaaaabaGaeyypa0ZaaeWa aeaadaaeqbqaamaaemaabaGaeqOXdO2aaSbaaSqaaiaad6gaaeqaaO WaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLhWUaayjcSdWaaWba aSqabeaacaaIYaaaaaqaaiaad6gaaeqaniabggHiLdaakiaawIcaca GLPaaadaqadaqaaiaaigdacqGHRaWkdaaeqbqaamaaemaabaGaeqOX dO2aaSbaaSqaaiaad2gaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaqaaiaad2ga aeqaniabggHiLdaakiaawIcacaGLPaaacqGH9aqpcaaIYaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGOaGaaeymaiaabodacaqGPaaaaaa@D281@ namely, it contains only two quanta. Where we utilize Eq.(8) and the following relation [24] is: n | ϕ n ( t ) | 2 =1, m | ϕ m ( t ) | 2 =1,[ B n . B m + ]= δ nm              (14) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaada abdaqaaiabew9aMnaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaa aaaeaacaWGUbaabeqdcqGHris5aOGaeyypa0JaaGymaiaacYcadaae qbqaamaaemaabaGaeqy1dy2aaSbaaSqaaiaad2gaaeqaaOWaaeWaae aacaWG0baacaGLOaGaayzkaaaacaGLhWUaayjcSdWaaWbaaSqabeaa caaIYaaaaaqaaiaad2gaaeqaniabggHiLdGccqGH9aqpcaaIXaGaai ilaiaacUfacaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaaiOlaiaadkea daqhaaWcbaGaamyBaaqaaiabgUcaRaaakiaac2facqGH9aqpcqaH0o azdaWgaaWcbaGaamOBaiaad2gaaeqaaOGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGXaGaaeinaiaabMcaaaa@6AFE@ ex <0| B n + |0 > ex = ex <0| B n + B n |0 > ex = ex <0| B n + B m |0 > ex        = ex <0| B n + B m B l |0 > ex = ex <0| B n + B m B l + B n |0 > ex        = ex <0| B n + B m B i + B l B j |0 > ex = ex <0| B n + B m B l + B i B j B n |0 > ex ....=0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWgaa WcbaGaamyzaiaadIhaaeqaaOGaeyipaWJaaGimaiaacYhacaWGcbWa a0baaSqaaiaad6gaaeaacqGHRaWkaaGccaGG8bGaaGimaiabg6da+m aaBaaaleaacaWGLbGaamiEaaqabaGccqGH9aqpdaWgaaWcbaGaamyz aiaadIhaaeqaaOGaeyipaWJaaGimaiaacYhacaWGcbWaa0baaSqaai aad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaad6gaaeqaaOGa aiiFaiaaicdacqGH+aGpdaWgaaWcbaGaamyzaiaadIhaaeqaaOGaey ypa0ZaaSbaaSqaaiaadwgacaWG4baabeaakiabgYda8iaaicdacaGG 8bGaamOqamaaDaaaleaacaWGUbaabaGaey4kaScaaOGaamOqamaaBa aaleaacaWGTbaabeaakiaacYhacaaIWaGaeyOpa4ZaaSbaaSqaaiaa dwgacaWG4baabeaaaOqaaiaabccacaqGGaGaaeiiaiaabccacaqGGa Gaaeiiaiabg2da9maaBaaaleaacaWGLbGaamiEaaqabaGccqGH8aap caaIWaGaaiiFaiaadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaaki aadkeadaWgaaWcbaGaamyBaaqabaGccaWGcbWaaSbaaSqaaiaadYga aeqaaOGaaiiFaiaaicdacqGH+aGpdaWgaaWcbaGaamyzaiaadIhaae qaaOGaeyypa0ZaaSbaaSqaaiaadwgacaWG4baabeaakiabgYda8iaa icdacaGG8bGaamOqamaaDaaaleaacaWGUbaabaGaey4kaScaaOGaam OqamaaBaaaleaacaWGTbaabeaakiaadkeadaqhaaWcbaGaamiBaaqa aiabgUcaRaaakiaadkeadaWgaaWcbaGaamOBaaqabaGccaGG8bGaaG imaiabg6da+maaBaaaleaacaWGLbGaamiEaaqabaaakeaacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacqGH9aqpdaWgaaWcbaGaam yzaiaadIhaaeqaaOGaeyipaWJaaGimaiaacYhacaWGcbWaa0baaSqa aiaad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqaaiaad2gaaeqaaO GaamOqamaaDaaaleaacaWGPbaabaGaey4kaScaaOGaamOqamaaBaaa leaacaWGSbaabeaakiaadkeadaWgaaWcbaGaamOAaaqabaGccaGG8b GaaGimaiabg6da+maaBaaaleaacaWGLbGaamiEaaqabaGccqGH9aqp daWgaaWcbaGaamyzaiaadIhaaeqaaOGaeyipaWJaaGimaiaacYhaca WGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaSqa aiaad2gaaeqaaOGaamOqamaaDaaaleaacaWGSbaabaGaey4kaScaaO GaamOqamaaBaaaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamOA aaqabaGccaWGcbWaaSbaaSqaaiaad6gaaeqaaOGaaiiFaiaaicdacq GH+aGpdaWgaaWcbaGaamyzaiaadIhaaeqaaOGaaiOlaiaac6cacaGG UaGaaiOlaiabg2da9iaaicdaaaaa@C1B2@ Therefore, the new wave function is completely different from Davydov’s. The latter is an excitation state of a single particle with one quantum and an eigenstate of the number operator, but the former is not. The new state is a quasicoherent state. It contains only two excitons, which come from the second and third terms in Eq.(9), in which each term contributes only an exciton, but it is not an excitation state of two single parties. Hence, as far as the form of new wave function in Eq.(9) is concerned, it is either two-quanta states proposed by Forner [21] and Cruzeiro- Hansson [10,18] or a standard coherent state proposed by Brown et al. [4,2] and Kerr et al’s [13] and Schweitzer et al’s 15,[21] multiquanta states. Therefore, the wave function, Eq. (9), is new for the protein molecular systems. It not only exhibits the coherent feature of the collective excitation of excitons and phonons caused by the nonlinear interaction generated by the exciton-phonon interaction, which , thus, also makes the wave function of the states of the system symmetrical, but it also agrees with the fact that the energy released in the ATP hydrolysis (about 0.43 eV) may only create two amide-I vibrational quanta which, thus, can also make the numbers of excitons maintain conservation in the Hamiltonian, Eq.(10). Meanwhile, the new wave function has another advantage, i.e., the equation of motion of the soliton can also be obtained from the Heisenberg equations of the creation and annihilation operators in quantum mechanics by using Eqs. (9) and (10), but the wave function of the states of the system in other models could not, including the one-quanta state [3] and the two-quanta state [12,22]. Therefore, the above Hamitonian and wave function, Eqs. (9) and (10), are reasonable and appropriate to the protein molecules.
The dynamic equation of Bio-energy transport
We now derive the equations of motion from Pang’s model. First of all, we give the interpretation of β n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaa@3B11@ and π n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaa@3B2D@ in Eq.(9).We know that the phonon part of the new wave function in Eq.(9) depending on the displacement and momentum operators is a coherent state of the normal model of creation and annihilation operators. A coherent state for the mode with wave vector q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyCaaaa@36E9@ is [3,12,24-26] |α(t) =exp( q [ α q (t) a q + α q * (t) a q ] ) |0 ph           (15) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaGaaeaaca GG8bGaeqySdeMaaiikaiaadshacaGGPaaacaGLQmcacqGH9aqpciGG LbGaaiiEaiaacchadaqadaqaamaaqafabaGaai4waiabeg7aHnaaBa aaleaacaWGXbaabeaakiaacIcacaWG0bGaaiykaiaadggadaqhaaWc baGaamyCaaqaaiabgUcaRaaakiabgkHiTiabeg7aHnaaDaaaleaaca WGXbaabaGaaiOkaaaakiaacIcacaWG0bGaaiykaiaadggadaWgaaWc baGaamyCaaqabaGccaGGDbaaleaacaWGXbaabeqdcqGHris5aaGcca GLOaGaayzkaaWaaaGaaeaacaGG8bGaaGimaaGaayPkJaWaaSbaaSqa aiaadchacaWGObaabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabwda caqGPaaaaa@64D9@ Utilizing the standard transformations u n = q [ 2NM ω q ] 1/2 e iqn r 0 ( a q + + a q ) , P n =i q [ M ω q 2N ] 1/2 e iqn r 0 ( a q + a q )   (16) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGUbaabeaakiabg2da9maaqafabaWaamWaaeaadaWcaaqa aiabl+qiObqaaiaaikdacaWGobGaamytaiabeM8a3naaBaaaleaaca WGXbaabeaaaaaakiaawUfacaGLDbaadaahaaWcbeqaamaalyaabaGa aGymaaqaaiaaikdaaaaaaOGaamyzamaaCaaaleqabaGaamyAaiaadg hacaWGUbGaamOCamaaBaaameaacaaIWaaabeaaaaGccaGGOaGaamyy amaaDaaaleaacqGHsislcaWGXbaabaGaey4kaScaaOGaey4kaSIaam yyamaaBaaaleaacaWGXbaabeaakiaacMcaaSqaaiaadghaaeqaniab ggHiLdGccaGGSaGaamiuamaaBaaaleaacaWGUbaabeaakiabg2da9i aadMgadaaeqbqaamaadmaabaWaaSaaaeaacaWGnbGaeS4dHGMaeqyY dC3aaSbaaSqaaiaadghaaeqaaaGcbaGaaGOmaiaad6eaaaaacaGLBb GaayzxaaWaaWbaaSqabeaadaWcgaqaaiaaigdaaeaacaaIYaaaaaaa kiaadwgadaahaaWcbeqaaiaadMgacaWGXbGaamOBaiaadkhadaWgaa adbaGaaGimaaqabaaaaOGaaiikaiaadggadaqhaaWcbaGaeyOeI0Ia amyCaaqaaiabgUcaRaaakiabgkHiTiaadggadaWgaaWcbaGaamyCaa qabaGccaGGPaaaleaacaWGXbaabeqdcqGHris5aOGaaeiiaiaabcca caqGOaGaaeymaiaabAdacaqGPaaaaa@78B7@ we can get [12,23] | α(t) = | β(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqqaaeaada aacaqaaiabeg7aHjaacIcacaWG0bGaaiykaaGaayPkJaGaeyypa0da caGLhWoadaaacaqaamaaeeaabaGaeqOSdiMaaiikaiaadshacaGGPa aacaGLhWoaaiaawQYiaaaa@43DD@ ,where | β(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaGaaeaada abbaqaaiabek7aIjaacIcacaWG0bGaaiykaaGaay5bSdaacaGLQmca aaa@3C67@ is in Eq.(9), and ω q =2 (w/M ) 1/2 sin( r 0 q /2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadghaaeqaaOGaeyypa0JaaGOmaiaacIcadaWcgaqaaiaa dEhaaeaacaWGnbaaaiaacMcadaahaaWcbeqaamaalyaabaGaaGymaa qaaiaaikdaaaaaaOGaci4CaiaacMgacaGGUbGaaiikamaalyaabaGa amOCamaaBaaaleaacaaIWaaabeaakiaadghaaeaacaaIYaaaaiaacM caaaa@4791@ is the distance between neighboring amino acid molecules, and a q ( a q + ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyyamaaBa aaleaacaqGXbaabeaakiaacIcacaqGHbWaa0baaSqaaiaabghaaeaa cqGHRaWkaaGccaGGPaaaaa@3C4D@ is the annihilation (creation) operator of the phonon with wave vector q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyCaaaa@36E9@ ,where α(t)| a q |α(t) = a q (t)= ( M ω q 2h ) 1/2 β q (t)+i ( 1 2M ω q ) 1/2 π q (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq aHXoqycaGGOaGaaeiDaiaacMcadaabdaqaaiaabggadaWgaaWcbaGa aeyCaaqabaaakiaawEa7caGLiWoacqaHXoqycaGGOaGaaeiDaiaacM caaiaawMYicaGLQmcacqGH9aqpcaqGHbWaaSbaaSqaaiaabghaaeqa aOGaaiikaiaabshacaGGPaGaeyypa0ZaaeWaaeaadaWcaaqaaiaab2 eacqaHjpWDdaWgaaWcbaGaaeyCaaqabaaakeaacaqGYaGaaeiAaaaa aiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaaeymaaqaaiaabk daaaaaaOGaeqOSdi2aaSbaaSqaaiaabghaaeqaaOGaaiikaiaabsha caGGPaGaey4kaSIaaeyAamaabmaabaWaaSaaaeaacaqGXaaabaGaae Omaiaab2eacqWIpecAcqaHjpWDdaWgaaWcbaGaaeyCaaqabaaaaaGc caGLOaGaayzkaaWaaWbaaSqabeaadaWcgaqaaiaabgdaaeaacaqGYa aaaaaakiabec8aWnaaBaaaleaacaqGXbaabeaakiaacIcacaqG0bGa aiykaaaa@69BB@ β q (t)= 1 N n e iqnr 0 π n (t) , π q (t)= 1 N n e iqnr 0 π n (t) , Φ(t)| P n |Φ(t) = π n (t)  (17) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaabghaaeqaaOGaaiikaiaabshacaGGPaGaeyypa0ZaaSaa aeaacaqGXaaabaWaaOaaaeaacaqGobaaleqaaaaakmaaqafabaGaae yzamaaCaaaleqabaGaeyOeI0IaaeyAaiaabghacaqGUbGaaeOCamaa BaaameaacaqGWaaabeaaaaGccqaHapaCdaWgaaWcbaGaaeOBaaqaba GccaGGOaGaaeiDaiaacMcaaSqaaiaab6gaaeqaniabggHiLdGccaGG SaGaeqiWda3aaSbaaSqaaiaabghaaeqaaOGaaiikaiaabshacaGGPa Gaeyypa0ZaaSaaaeaacaqGXaaabaWaaOaaaeaacaqGobaaleqaaaaa kmaaqafabaGaaeyzamaaCaaaleqabaGaeyOeI0IaaeyAaiaabghaca qGUbGaaeOCamaaBaaameaacaqGWaaabeaaaaGccqaHapaCdaWgaaWc baGaaeOBaaqabaGccaGGOaGaaeiDaiaacMcaaSqaaiaab6gaaeqani abggHiLdGccaGGSaWaaaWaaeaacqqHMoGrcaGGOaGaaeiDaiaacMca daabdaqaaiaabcfadaWgaaWcbaGaaeOBaaqabaaakiaawEa7caGLiW oacqqHMoGrcaGGOaGaaeiDaiaacMcaaiaawMYicaGLQmcacqGH9aqp cqaHapaCdaWgaaWcbaGaaeOBaaqabaGccaGGOaGaaeiDaiaacMcaca qGGaGaaeiiaiaabIcacaqGXaGaae4naiaabMcaaaa@7DF4@ Utilizing again the above results and the formulas of the expectation values of the Heisenberg equations of operators u n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGUbaabeaaaaa@380E@ and P n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGUbaabeaaaaa@37E9@ , in the state | Φ(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaGaaeaada abbaqaaiabfA6agjaacIcacaWG0bGaaiykaaGaay5bSdaacaGLQmca aaa@3C40@ . i t Φ(t)| u n |Φ(t) = Φ(t)| [ u n ,H ] |Φ(t) ,i t Φ(t)| P n |Φ(t) = Φ(t)| [ u n ,H ] |Φ(t)   (18) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabl+ qiOnaalaaabaGaeyOaIylabaGaeyOaIyRaamiDaaaadaaadaqaaiab fA6agjaacIcacaqG0bGaaiykamaaemaabaGaaeyDamaaBaaaleaaca qGUbaabeaaaOGaay5bSlaawIa7aiabfA6agjaacIcacaqG0bGaaiyk aaGaayzkJiaawQYiaiabg2da9maaamaabaGaeuOPdyKaaiikaiaabs hacaGGPaWaaqWaaeaadaWadaqaaiaabwhadaWgaaWcbaGaaeOBaaqa baGccaGGSaGaamisaaGaay5waiaaw2faaaGaay5bSlaawIa7aiabfA 6agjaacIcacaqG0bGaaiykaaGaayzkJiaawQYiaiaacYcacaWGPbGa eS4dHG2aaSaaaeaacqGHciITaeaacqGHciITcaWG0baaamaaamaaba GaeuOPdyKaaiikaiaabshacaGGPaWaaqWaaeaacaqGqbWaaSbaaSqa aiaab6gaaeqaaaGccaGLhWUaayjcSdGaeuOPdyKaaiikaiaabshaca GGPaaacaGLPmIaayPkJaGaeyypa0ZaaaWaaeaacqqHMoGrcaGGOaGa aeiDaiaacMcadaabdaqaamaadmaabaGaaeyDamaaBaaaleaacaqGUb aabeaakiaacYcacaWGibaacaGLBbGaayzxaaaacaGLhWUaayjcSdGa euOPdyKaaiikaiaabshacaGGPaaacaGLPmIaayPkJaGaaeiiaiaabc cacaqGOaGaaeymaiaabIdacaqGPaaaaa@89E2@ We can obtain the equation of motion for the β n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaa@3B11@ as M β ¨ n (t)=w[ β n+1 (t)2 β n (t)+ β n1 (t) ]+2 χ 1 [ | ϕ n+1 (t) | 2 | ϕ n1 (t) | 2 ]+ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiqbek 7aIzaadaWaaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaGa eyypa0Jaam4DamaadmaabaGaeqOSdi2aaSbaaSqaaiaad6gacqGHRa WkcaaIXaaabeaakiaacIcacaWG0bGaaiykaiabgkHiTiaaikdacqaH YoGydaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaacMcacqGHRa WkcqaHYoGydaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaaiik aiaadshacaGGPaaacaGLBbGaayzxaaGaey4kaSIaaGOmaiabeE8aJn aaBaaaleaacaaIXaaabeaakmaadeaabaWaaqWaaeaacqaHvpGzdaWg aaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaadshacaGGPa aacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaGccaGLBbaadaWa caqaaiabgkHiTmaaemaabaGaeqy1dy2aaSbaaSqaaiaad6gacqGHsi slcaaIXaaabeaakiaacIcacaWG0bGaaiykaaGaay5bSlaawIa7amaa CaaaleqabaGaaGOmaaaaaOGaayzxaaGaey4kaScaaa@7308@ 2 χ 2 { ϕ n * (t)[ ϕ n+1 (t) ϕ n1 (t) ] + ϕ n (t)[ ϕ n+1 * (t) ϕ n1 * (t) ] }   (19) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeE 8aJnaaBaaaleaacaaIYaaabeaakmaaceaabaGaeqy1dy2aa0baaSqa aiaad6gaaeaacaGGQaaaaOGaaiikaiaadshacaGGPaWaamWaaeaacq aHvpGzdaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaa dshacaGGPaGaeyOeI0Iaeqy1dy2aaSbaaSqaaiaad6gacqGHsislca aIXaaabeaakiaacIcacaWG0bGaaiykaaGaay5waiaaw2faaaGaay5E aaGaey4kaSYaaiGaaeaacqaHvpGzdaWgaaWcbaGaamOBaaqabaGcca GGOaGaamiDaiaacMcadaWadaqaaiabew9aMnaaDaaaleaacaWGUbGa ey4kaSIaaGymaaqaaiaacQcaaaGccaGGOaGaamiDaiaacMcacqGHsi slcqaHvpGzdaqhaaWcbaGaamOBaiabgkHiTiaaigdaaeaacaGGQaaa aOGaaiikaiaadshacaGGPaaacaGLBbGaayzxaaaacaGL9baacaqGGa GaaeiiaiaabccacaqGOaGaaeymaiaabMdacaqGPaaaaa@6F0A@ From Eq.(19) we see that the presence of two quanta for the oscillators increases the driving force on the phonon field by that factor, when compared with the Davydov theory.

We now derive the equation of motion for the φ n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aaS baaSqaaiaad6gaaeqaaaaa@38D1@ . A basic assumption in the derivation is that | Φ(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaGaaeaada abbaqaaiabfA6agjaacIcacaWG0bGaaiykaaGaay5bSdaacaGLQmca aaa@3C40@ in Eq. (9) is a solution of the time-dependent equation [24-26]: i t | Φ(t) =H | Φ(t)      (20) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyAaiabl+ qiOnaalaaabaGaeyOaIylabaGaeyOaIyRaaeiDaaaadaaacaqaamaa eeaabaGaeuOPdyKaaiikaiaabshacaGGPaaacaGLhWoaaiaawQYiai abg2da9iaabIeadaaacaqaamaaeeaabaGaeuOPdyKaaiikaiaabsha caGGPaaacaGLhWoaaiaawQYiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeikaiaabkdacaqGWaGaaeykaaaa@502E@ The left-hand side of Eq. (16) has [12,23] i t | Φ(t) ={ i( n ϕ ˙ n (t) B n + + n ϕ ˙ n (t) ϕ n (t) B n + B n + |0 ex ) } | β(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabl+ qiOnaalaaabaGaeyOaIylabaGaeyOaIyRaamiDaaaadaaacaqaamaa eeaabaGaeuOPdyKaaiikaiaadshacaGGPaaacaGLhWoaaiaawQYiai abg2da9maacmaabaGaamyAaiabl+qiOnaabmaabaWaaabuaeaacuaH vpGzgaGaamaaBaaaleaacaWGUbaabeaakiaacIcacaWG0bGaaiykai aadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiabgUcaRmaaqafa baGafqy1dyMbaiaadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDai aacMcacqaHvpGzdaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaa cMcacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaa0 baaSqaaiaad6gaaeaacqGHRaWkaaGcdaaacaqaamaaeeaabaGaaGim aaGaay5bSdaacaGLQmcadaWgaaWcbaGaamyzaiaadIhaaeqaaaqaai aad6gaaeqaniabggHiLdaaleaacaWGUbaabeqdcqGHris5aaGccaGL OaGaayzkaaaacaGL7bGaayzFaaWaaaGaaeaadaabbaqaaiabek7aIj aacIcacaWG0bGaaiykaaGaay5bSdaacaGLQmcaaaa@7422@ + | φ P (t) { n { β n (t) P n π n (t) u n + 1 2 [ β n (t) π ˙ n (t) β ˙ n (t) π n (t) ] } | β(t) }   (21) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSYaaa GaaeaadaabbaqaaiabeA8aQnaaBaaaleaacaWGqbaabeaakiaacIca caWG0bGaaiykaaGaay5bSdaacaGLQmcadaGadaqaamaaqafabaWaai WaaeaacqaHYoGydaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaa cMcacaWGqbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaeqiWda3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaGaamyDamaaBaaa leaacaWGUbaabeaakiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaa WaamWaaeaacqaHYoGydaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiD aiaacMcacuaHapaCgaGaamaaBaaaleaacaWGUbaabeaakiaacIcaca WG0bGaaiykaiabgkHiTiqbek7aIzaacaWaaSbaaSqaaiaad6gaaeqa aOGaaiikaiaadshacaGGPaGaeqiWda3aaSbaaSqaaiaad6gaaeqaaO GaaiikaiaadshacaGGPaaacaGLBbGaayzxaaaacaGL7bGaayzFaaWa aaGaaeaadaabbaqaaiabek7aIjaacIcacaWG0bGaaiykaaGaay5bSd aacaGLQmcaaSqaaiaad6gaaeqaniabggHiLdaakiaawUhacaGL9baa caqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabgdacaqGPaaaaa@7A71@ Now left-multiplying the both sides of Eq.(21) by Φ(t) | MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaqaaeaada abcaqaaiabfA6agjaacIcacaqG0bGaaiykaaGaayjcSdaacaGLPmca aaa@3C3E@ ,the lefthand side of Eq.(21) can be i Φ(t)| u n |Φ(t) i n ϕ n * (t) ϕ n (t) ( m ϕ m * (t) ϕ m (t)+1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyAaiabl+ qiOnaaamaabaGaeuOPdyKaaiikaiaabshacaGGPaWaaqWaaeaacaqG 1bWaaSbaaSqaaiaab6gaaeqaaaGccaGLhWUaayjcSdGaeuOPdyKaai ikaiaabshacaGGPaaacaGLPmIaayPkJaGaaeyAaiabl+qiOnaaqafa baGaeqy1dy2aa0baaSqaaiaab6gaaeaacaGGQaaaaOGaaiikaiaabs hacaGGPaGaeqy1dy2aaSbaaSqaaiaab6gaaeqaaOGaaiikaiaabsha caGGPaaaleaacaqGUbaabeqdcqGHris5aOWaaeWaaeaadaaeqbqaai abew9aMnaaDaaaleaacaqGTbaabaGaaiOkaaaakiaacIcacaqG0bGa aiykaiabew9aMnaaBaaaleaacaqGTbaabeaakiaacIcacaqG0bGaai ykaiabgUcaRiaabgdaaSqaaiaab2gaaeqaniabggHiLdaakiaawIca caGLPaaaaaa@6876@ = 5 4 n [ β ˙ n (t) π n (t) π ˙ n (t) β n (t) ] n | ϕ n (t) | 2     (22) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaqG1aaabaGaaeinaaaadaaeqbqaamaadmaabaGafqOSdiMb aiaadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaacMcacqaHap aCdaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaacMcacqGHsisl cuaHapaCgaGaamaaBaaaleaacaWGUbaabeaakiaacIcacaWG0bGaai ykaiabek7aInaaBaaaleaacaWGUbaabeaakiaacIcacaWG0bGaaiyk aaGaay5waiaaw2faaaWcbaGaaeOBaaqab0GaeyyeIuoakmaaqafaba WaaqWaaeaacqaHvpGzdaWgaaWcbaGaaeOBaaqabaGccaGGOaGaaeiD aiaacMcaaiaawEa7caGLiWoadaahaaWcbeqaaiaabkdaaaaabaGaae OBaaqab0GaeyyeIuoakiaabccacaqGGaGaaeiiaiaabccacaqGOaGa aeOmaiaabkdacaqGPaaaaa@64D4@ Similarly, for the right-hand side of Eq. (20) we can have [12,23] Φ(t)| ( H ex + H ph + H int ) |Φ(t) ={ n { ε 0 | ϕ n (t) 2 |J ϕ n * (t)[ ϕ n+1 (t) ϕ n1 (t) ] } ×( 1+ m | ϕ m (t) | 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq qHMoGrcaGGOaGaaeiDaiaacMcadaabdaqaamaabmaabaGaaeisamaa BaaaleaacaqGLbGaaeiEaaqabaGccqGHRaWkcaqGibWaaSbaaSqaai aabchacaqGObaabeaakiabgUcaRiaabIeadaWgaaWcbaGaaeyAaiaa b6gacaqG0baabeaaaOGaayjkaiaawMcaaaGaay5bSlaawIa7aiabfA 6agjaacIcacaqG0bGaaiykaaGaayzkJiaawQYiaiabg2da9maaceaa baWaaabuaeaadaGadaqaaiabew7aLnaaBaaaleaacaqGWaaabeaakm aaemaabaGaeqy1dy2aaSbaaSqaaiaab6gaaeqaaOGaaiikaiaabsha caGGPaWaaWbaaSqabeaacaqGYaaaaaGccaGLhWUaayjcSdGaeyOeI0 IaaeOsaiabew9aMnaaDaaaleaacaqGUbaabaGaaiOkaaaakiaacIca caqG0bGaaiykamaadmaabaGaeqy1dy2aaSbaaSqaaiaab6gacqGHRa WkcaqGXaaabeaakiaacIcacaqG0bGaaiykaiabgkHiTiabew9aMnaa BaaaleaacaqGUbGaeyOeI0IaaeymaaqabaGccaGGOaGaaeiDaiaacM caaiaawUfacaGLDbaaaiaawUhacaGL9baaaSqaaiaab6gaaeqaniab ggHiLdaakiaawUhaaiabgEna0oaabmaabaGaaGymaiabgUcaRmaaqa fabaWaaqWaaeaacqaHvpGzdaWgaaWcbaGaamyBaaqabaGccaGGOaGa amiDaiaacMcaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaaba GaamyBaaqab0GaeyyeIuoaaOGaayjkaiaawMcaaaaa@8C86@ +{ n { χ 1 [ β n+1 (t) β n1 (t) ] | ϕ n (t) | 2 + χ 2 [ β n+1 (t) β n1 (t) ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSYaai qaaeaadaaeqbqaamaaceaabaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqa aOWaamWaaeaacqaHYoGydaWgaaWcbaGaamOBaiabgUcaRiaaigdaae qaaOGaaiikaiaadshacaGGPaGaeyOeI0IaeqOSdi2aaSbaaSqaaiaa d6gacqGHsislcaaIXaaabeaakiaacIcacaWG0bGaaiykaaGaay5wai aaw2faamaaemaabaGaeqy1dy2aaSbaaSqaaiaad6gaaeqaaOGaaiik aiaadshacaGGPaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOWaamWaaeaacqaH YoGydaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaads hacaGGPaGaeyOeI0IaeqOSdi2aaSbaaSqaaiaad6gacqGHsislcaaI XaaabeaakiaacIcacaWG0bGaaiykaaGaay5waiaaw2faaaGaay5Eaa aaleaacaWGUbaabeqdcqGHris5aaGccaGL7baaaaa@6C6E@ × φ n * (t)[ φ n+1 (t) φ n1 (t) ] } }( 1+ m | φ m (t) | 2 )+ 5 2 W(t) n | φ n (t) | 2    (23) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada GacaqaaiabgEna0kabeA8aQnaaDaaaleaacaWGUbaabaGaaiOkaaaa kiaacIcacaWG0bGaaiykamaadmaabaGaeqOXdO2aaSbaaSqaaiaad6 gacqGHRaWkcaaIXaaabeaakiaacIcacaWG0bGaaiykaiabgkHiTiab eA8aQnaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaGGOaGaam iDaiaacMcaaiaawUfacaGLDbaaaiaaw2haaaGaayzFaaWaaeWaaeaa caaIXaGaey4kaSYaaabuaeaadaabdaqaaiabeA8aQnaaBaaaleaaca WGTbaabeaakiaacIcacaWG0bGaaiykaaGaay5bSlaawIa7amaaCaaa leqabaGaaGOmaaaaaeaacaWGTbaabeqdcqGHris5aaGccaGLOaGaay zkaaGaey4kaSYaaSaaaeaacaaI1aaabaGaaGOmaaaacaWGxbGaaiik aiaadshacaGGPaWaaabuaeaadaabdaqaaiabeA8aQnaaBaaaleaaca WGUbaabeaakiaacIcacaWG0bGaaiykaaGaay5bSlaawIa7amaaCaaa leqabaGaaGOmaaaaaeaacaWGUbaabeqdcqGHris5aOGaaeiiaiaabc cacaqGGaGaaeikaiaabkdacaqGZaGaaeykaaaa@76DB@ where W(t)= β(t)| H ph |β(t) = n ( 1 2M π n 2 (t)+ 1 2 w [ β n (t) β n1 (t) ] 2 ) + q 1 2 ω q   (24) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiaacI cacaWG0bGaaiykaiabg2da9maaamaabaGaeqOSdiMaaiikaiaabsha caGGPaWaaqWaaeaacaWGibWaaSbaaSqaaiaadchacaWGObaabeaaaO Gaay5bSlaawIa7aiabek7aIjaacIcacaqG0bGaaiykaaGaayzkJiaa wQYiaiabg2da9maaqafabaWaaeWaaeaadaWcaaqaaiaaigdaaeaaca aIYaGaamytaaaacqaHapaCdaqhaaWcbaGaamOBaaqaaiaaikdaaaGc caGGOaGaamiDaiaacMcacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYa aaaiaadEhadaWadaqaaiabek7aInaaBaaaleaacaWGUbaabeaakiaa cIcacaWG0bGaaiykaiabgkHiTiabek7aInaaBaaaleaacaWGUbGaey OeI0IaaGymaaqabaGccaGGOaGaamiDaiaacMcaaiaawUfacaGLDbaa daahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaSqaaiaad6gaae qaniabggHiLdGccqGHRaWkdaaeqbqaamaalaaabaGaaGymaaqaaiaa ikdaaaGaeS4dHGMaeqyYdC3aaSbaaSqaaiaadghaaeqaaaqaaiaadg haaeqaniabggHiLdGccaqGGaGaaeiiaiaabIcacaqGYaGaaeinaiaa bMcaaaa@7898@ and utilizing Eqs.(8) and (12)-(14) and the relationships can be obtained: n [ β m+1 (t)2 β m (t)+ β m1 (t) ] β m (t) = n [ β m+1 (t) β m1 (t) ] 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaada Wadaqaaiabek7aInaaBaaaleaacaqGTbGaey4kaSIaaGymaaqabaGc caGGOaGaamiDaiaacMcacqGHsislcaaIYaGaeqOSdi2aaSbaaSqaai aab2gaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaeqOSdi2aaSba aSqaaiaab2gacqGHsislcaaIXaaabeaakiaacIcacaWG0bGaaiykaa Gaay5waiaaw2faaiabek7aInaaBaaaleaacaWGTbaabeaakiaacIca caWG0bGaaiykaaWcbaGaaeOBaaqab0GaeyyeIuoakiabg2da9iabgk HiTmaaqafabaWaamWaaeaacqaHYoGydaWgaaWcbaGaaeyBaiabgUca RiaaigdaaeqaaOGaaiikaiaadshacaGGPaGaeyOeI0IaeqOSdi2aaS baaSqaaiaab2gacqGHsislcaaIXaaabeaakiaacIcacaWG0bGaaiyk aaGaay5waiaaw2faamaaCaaaleqabaGaaGOmaaaaaeaacaqGUbaabe qdcqGHris5aaaa@6B76@ Φ(t)| n ( B n + B n1 + B n B n1 + ) |Φ(t) = n [ ϕ n * (t) ϕ n+1 (t)+ ϕ n1 * (t)ϕ(t) ]( 1+ m | ϕ m (t) | 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq qHMoGrcaGGOaGaaeiDaiaacMcadaabdaqaamaaqafabaWaaeWaaeaa caWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaaSbaaS qaaiaad6gacqGHsislcaaIXaaabeaakiabgUcaRiaadkeadaWgaaWc baGaamOBaaqabaGccaWGcbWaa0baaSqaaiaad6gacqGHsislcaaIXa aabaGaey4kaScaaaGccaGLOaGaayzkaaaaleaacaWGUbaabeqdcqGH ris5aaGccaGLhWUaayjcSdGaeuOPdyKaaiikaiaabshacaGGPaaaca GLPmIaayPkJaGaeyypa0ZaaabuaeaadaWadaqaaiabew9aMnaaDaaa leaacaWGUbaabaGaaiOkaaaakiaacIcacaWG0bGaaiykaiabew9aMn aaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGOaGaamiDaiaa cMcacqGHRaWkcqaHvpGzdaqhaaWcbaGaamOBaiabgkHiTiaaigdaae aacaGGQaaaaOGaaiikaiaadshacaGGPaGaeqy1dyMaaiikaiaadsha caGGPaaacaGLBbGaayzxaaWaaeWaaeaacaaIXaGaey4kaSYaaabuae aadaabdaqaaiabew9aMnaaBaaaleaacaWGTbaabeaakiaacIcacaWG 0bGaaiykaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaeaaca WGTbaabeqdcqGHris5aaGccaGLOaGaayzkaaaaleaacaWGUbaabeqd cqGHris5aaaa@83D9@ Φ(t)| n ( u n+1 u n1 )( B n + B n ) |Φ(t) = n { [ β m+1 (t) β m1 (t) ] | ϕ n (t) | 2 }( 1+ m | ϕ m (t) | 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq qHMoGrcaGGOaGaaeiDaiaacMcadaabdaqaamaaqafabaWaaeWaaeaa caWG1bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTi aadwhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaGccaGLOaGa ayzkaaWaaeWaaeaacaWGcbWaa0baaSqaaiaad6gaaeaacqGHRaWkaa GccaWGcbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaleaa caWGUbaabeqdcqGHris5aaGccaGLhWUaayjcSdGaeuOPdyKaaiikai aabshacaGGPaaacaGLPmIaayPkJaGaeyypa0ZaaabuaeaadaGadaqa amaadmaabaGaeqOSdi2aaSbaaSqaaiaab2gacqGHRaWkcaaIXaaabe aakiaacIcacaWG0bGaaiykaiabgkHiTiabek7aInaaBaaaleaacaqG TbGaeyOeI0IaaGymaaqabaGccaGGOaGaamiDaiaacMcaaiaawUfaca GLDbaadaabdaqaaiabew9aMnaaBaaaleaacaWGUbaabeaakiaacIca caWG0bGaaiykaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaO Gaay5Eaiaaw2haamaabmaabaGaaGymaiabgUcaRmaaqafabaWaaqWa aeaacqaHvpGzdaWgaaWcbaGaamyBaaqabaGccaGGOaGaamiDaiaacM caaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaabaGaamyBaaqa b0GaeyyeIuoaaOGaayjkaiaawMcaaaWcbaGaamOBaaqab0GaeyyeIu oaaaa@856A@ Φ(t)| n ( u n1 u n )( B n + B n1 + B n B n1 + ) |Φ(t) = n { [ β m+1 (t) β m1 (t) ][ ϕ n * (t) ϕ n+1 (t)+ ϕ n1 * (t)ϕ(t) ] } ×( 1+ m | ϕ m (t) | 2 )  (25) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaaada qaaiabfA6agjaacIcacaqG0bGaaiykamaaemaabaWaaabuaeaadaqa daqaaiaadwhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaey OeI0IaamyDamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaa bmaabaGaamOqamaaDaaaleaacaWGUbaabaGaey4kaScaaOGaamOqam aaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHRaWkcaWGcbWa aSbaaSqaaiaad6gaaeqaaOGaamOqamaaDaaaleaacaWGUbGaeyOeI0 IaaGymaaqaaiabgUcaRaaaaOGaayjkaiaawMcaaaWcbaGaamOBaaqa b0GaeyyeIuoaaOGaay5bSlaawIa7aiabfA6agjaacIcacaqG0bGaai ykaaGaayzkJiaawQYiaiabg2da9aqaamaaqafabaWaaiWaaeaadaWa daqaaiabek7aInaaBaaaleaacaqGTbGaey4kaSIaaGymaaqabaGcca GGOaGaamiDaiaacMcacqGHsislcqaHYoGydaWgaaWcbaGaaeyBaiab gkHiTiaaigdaaeqaaOGaaiikaiaadshacaGGPaaacaGLBbGaayzxaa WaamWaaeaacqaHvpGzdaqhaaWcbaGaamOBaaqaaiaacQcaaaGccaGG OaGaamiDaiaacMcacqaHvpGzdaWgaaWcbaGaamOBaiabgUcaRiaaig daaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaeqy1dy2aa0baaSqa aiaad6gacqGHsislcaaIXaaabaGaaiOkaaaakiaacIcacaWG0bGaai ykaiabew9aMjaacIcacaWG0bGaaiykaaGaay5waiaaw2faaaGaay5E aiaaw2haaaWcbaGaamOBaaqab0GaeyyeIuoakiabgEna0oaabmaaba GaaGymaiabgUcaRmaaqafabaWaaqWaaeaacqaHvpGzdaWgaaWcbaGa amyBaaqabaGccaGGOaGaamiDaiaacMcaaiaawEa7caGLiWoadaahaa WcbeqaaiaaikdaaaaabaGaamyBaaqab0GaeyyeIuoaaOGaayjkaiaa wMcaaiaabccacaqGGaGaaeikaiaabkdacaqG1aGaaeykaaaaaa@A4F6@ From Eqs.(20)-(23) we can obtain i t ϕ n (t)= ε 0 ϕ n (t)J[ ϕ n+1 (t)+ ϕ n1 (t) ]+ χ 1 [ β n+1 (t)+ β n1 (t) ] ϕ n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabl+ qiOnaalaaabaGaeyOaIylabaGaeyOaIyRaamiDaaaacqaHvpGzdaWg aaWcbaGaamOBaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcqaH1o qzdaWgaaWcbaGaaGimaaqabaGccqaHvpGzdaWgaaWcbaGaamOBaaqa baGccaGGOaGaamiDaiaacMcacqGHsislcaWGkbWaamWaaeaacqaHvp GzdaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaadsha caGGPaGaey4kaSIaeqy1dy2aaSbaaSqaaiaad6gacqGHsislcaaIXa aabeaakiaacIcacaWG0bGaaiykaaGaay5waiaaw2faaiabgUcaRiab eE8aJnaaBaaaleaacaaIXaaabeaakmaadmaabaGaeqOSdi2aaSbaaS qaaiaad6gacqGHRaWkcaaIXaaabeaakiaacIcacaWG0bGaaiykaiab gUcaRiabek7aInaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcca GGOaGaamiDaiaacMcaaiaawUfacaGLDbaacqaHvpGzdaWgaaWcbaGa amOBaaqabaGccaGGOaGaamiDaiaacMcaaaa@757D@ χ 2 [ β n+1 (t)+ β n (t) ]×[ ϕ n+1 (t)+ ϕ n1 (t) ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq 4Xdm2aaSbaaSqaaiaaikdaaeqaaOWaamWaaeaacqaHYoGydaWgaaWc baGaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaadshacaGGPaGaey 4kaSIaeqOSdi2aaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGG PaaacaGLBbGaayzxaaGaey41aq7aamWaaeaacqaHvpGzdaWgaaWcba GaamOBaiabgUcaRiaaigdaaeqaaOGaaiikaiaadshacaGGPaGaey4k aSIaeqy1dy2aaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaacI cacaWG0bGaaiykaaGaay5waiaaw2faaaaa@5AEA@ + 5 2 ( W(t) 1 2 m [ β ˙ m (t) π m (t) π ˙ m (t)β(t) ] ) ϕ n (t)    (26) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSYaaS aaaeaacaaI1aaabaGaaGOmaaaadaqadaqaaiaadEfacaGGOaGaamiD aiaacMcacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaaqafaba WaamWaaeaacuaHYoGygaGaamaaBaaaleaacaWGTbaabeaakiaacIca caWG0bGaaiykaiabec8aWnaaBaaaleaacaWGTbaabeaakiaacIcaca WG0bGaaiykaiabgkHiTiqbec8aWzaacaWaaSbaaSqaaiaad2gaaeqa aOGaaiikaiaadshacaGGPaGaeqOSdiMaaiikaiaadshacaGGPaaaca GLBbGaayzxaaaaleaacaWGTbaabeqdcqGHris5aaGccaGLOaGaayzk aaGaeqy1dy2aaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeOnaiaabMca aaa@63AD@ In the continuum approximation we get from Eqs.(19) and (26) i t ϕ(x,t)=R(t)ϕ(x,t)J r 0 2 2 x 2 ϕ(x,t) G p | ϕ(x,t) | 2 ϕ(x,t)   (27) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabl+ qiOnaalaaabaGaeyOaIylabaGaeyOaIyRaamiDaaaacqaHvpGzcaGG OaGaamiEaiaacYcacaWG0bGaaiykaiabg2da9iaadkfacaGGOaGaam iDaiaacMcacqaHvpGzcaGGOaGaamiEaiaacYcacaWG0bGaaiykaiab gkHiTiaadQeacaWGYbWaa0baaSqaaiaaicdaaeaacaaIYaaaaOWaaS aaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaaakeaacqGHciITcaWG 4bWaaWbaaSqabeaacaaIYaaaaaaakiabew9aMjaacIcacaWG4bGaai ilaiaadshacaGGPaGaeyOeI0Iaam4ramaaBaaaleaacaWGWbaabeaa kmaaemaabaGaeqy1dyMaaiikaiaadIhacaGGSaGaamiDaiaacMcaai aawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqaHvpGzcaGGOaGa amiEaiaacYcacaWG0bGaaiykaiaabccacaqGGaGaaeiiaiaabIcaca qGYaGaae4naiaabMcaaaa@72BA@ and β(x,t) ξ = β(x,t) x = 4( χ 1 + χ 1 ) w( 1 s 2 ) r 0 | ϕ(x,t) | 2    (28) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcqaHYoGycaGGOaGaamiEaiaacYcacaWG0bGaaiykaaqaaiab gkGi2kabe67a4baacqGH9aqpdaWcaaqaaiabgkGi2kabek7aIjaacI cacaWG4bGaaiilaiaadshacaGGPaaabaGaeyOaIyRaamiEaaaacqGH 9aqpcqGHsisldaWcaaqaaiaaisdadaqadaqaaiabeE8aJnaaBaaale aacaaIXaaabeaakiabgUcaRiabeE8aJnaaBaaaleaacaaIXaaabeaa aOGaayjkaiaawMcaaaqaaiaadEhadaqadaqaaiaaigdacqGHsislca WGZbWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaamOCamaa BaaaleaacaaIWaaabeaaaaGcdaabdaqaaiabew9aMjaacIcacaWG4b GaaiilaiaadshacaGGPaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaI YaaaaOGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG4aGaaeykaa aa@6BBA@ here ξ=x v t MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaey ypa0JaamiEaiabgkHiTiaadAhadaWgaaWcbaGaamiDaaqabaaaaa@3CC8@   R(t)= ε 0 2J+ 5 2 { W(t) 1 2 m [ β ˙ m (t) π m (t) π ˙ m (t)β(t) ] } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaabc cacaWGsbGaaiikaiaadshacaGGPaGaeyypa0JaeqyTdu2aaSbaaSqa aiaaicdaaeqaaOGaeyOeI0IaaGOmaiaadQeacqGHRaWkdaWcaaqaai aaiwdaaeaacaaIYaaaamaacmaabaGaam4vaiaacIcacaWG0bGaaiyk aiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaWaaabuaeaadaWada qaaiqbek7aIzaacaWaaSbaaSqaaiaad2gaaeqaaOGaaiikaiaadsha caGGPaGaeqiWda3aaSbaaSqaaiaad2gaaeqaaOGaaiikaiaadshaca GGPaGaeyOeI0IafqiWdaNbaiaadaWgaaWcbaGaamyBaaqabaGccaGG OaGaamiDaiaacMcacqaHYoGycaGGOaGaamiDaiaacMcaaiaawUfaca GLDbaaaSqaaiaad2gaaeqaniabggHiLdaakiaawUhacaGL9baaaaa@6445@ and s=v/ v 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9maalyaabaGaamODaaqaaiaadAhadaWgbaWcbaGaaGimaaqabaaa aaaa@3AE6@ . The soliton solution of Eq.(27) is thus ϕ(x,t)= ( μ p 2 ) 1/2 sech[ ( μ p / r 0 )( x x 0 V t ) ]×exp{ i[ v 2J r 0 2 ( x x 0 ) E v t ] }   (29) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGSaGaamiDaiaacMcacqGH9aqpdaqadaqaamaalaaa baGaeqiVd02aaSbaaSqaaiaadchaaeqaaaGcbaGaaGOmaaaaaiaawI cacaGLPaaadaahaaWcbeqaamaalyaabaGaaGymaaqaaiaaikdaaaaa aOGaci4CaiaacwgacaGGJbGaamiAamaadmaabaWaaeWaaeaadaWcga qaaiabeY7aTnaaBaaaleaacaWGWbaabeaaaOqaaiaadkhadaWgaaWc baGaaGimaaqabaaaaaGccaGLOaGaayzkaaWaaeWaaeaacaWG4bGaey OeI0IaamiEamaaBaaaleaacaaIWaaabeaakiabgkHiTiaadAfadaWg aaWcbaGaamiDaaqabaaakiaawIcacaGLPaaaaiaawUfacaGLDbaacq GHxdaTciGGLbGaaiiEaiaacchadaGadaqaaiaadMgadaWadaqaamaa laaabaGaeS4dHGMaamODaaqaaiaaikdacaWGkbGaamOCamaaDaaale aacaaIWaaabaGaaGOmaaaaaaGcdaqadaqaaiaadIhacqGHsislcaWG 4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0Iaam yramaaBaaaleaacaWG2baabeaakmaalaaabaGaamiDaaqaaiabl+qi ObaaaiaawUfacaGLDbaaaiaawUhacaGL9baacaqGGaGaaeiiaiaabc cacaqGOaGaaeOmaiaabMdacaqGPaaaaa@77B3@ with μ P = 2 ( χ 1 + χ 2 ) 2 w( 1 s 2 )J , G P = 8 ( χ 1 + χ 2 ) 2 w( 1 s 2 )     (30) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadcfaaeqaaOGaeyypa0ZaaSaaaeaacaaIYaWaaeWaaeaa cqaHhpWydaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHhpWydaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikda aaaakeaacaWG3bWaaeWaaeaacaaIXaGaeyOeI0Iaam4CamaaCaaale qabaGaaGOmaaaaaOGaayjkaiaawMcaaiaadQeaaaGaaiilaiaadEea daWgaaWcbaGaamiuaaqabaGccqGH9aqpdaWcaaqaaiaaiIdadaqada qaaiabeE8aJnaaBaaaleaacaaIXaaabeaakiabgUcaRiabeE8aJnaa BaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaG OmaaaaaOqaaiaadEhadaqadaqaaiaaigdacqGHsislcaWGZbWaaWba aSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaiaabccacaqGGaGaae iiaiaabccacaqGOaGaae4maiaabcdacaqGPaaaaa@6290@ Although forms of the above equations of motion and the corresponding solution, Eqs.(27)- (30), are quite similar to that of the Davydov soliton, the properties of new soliton have very large differences from the latter because the parameter values in the equation of motion and the solution Eqs.(27) and (29), including R(t), G P , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaacI cacaWG0bGaaiykaiaacYcacaWGhbWaaSbaaSqaaiaadcfaaeqaaOGa aiilaaaa@3C55@ and μ P MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadcfaaeqaaaaa@38AC@ have obvious distinctions from that in the Davydov model. A straightforward result of Pang’s model is to increase the nonlinear interaction energy G P ( G P =2 G D [ 1+2( χ 2 / χ 1 )+ ( χ 2 / χ 1 ) 2 ] ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGqbaabeaakmaabmaabaGaam4ramaaBaaaleaacaWGqbaa beaakiabg2da9iaaikdacaWGhbWaaSbaaSqaaiaadseaaeqaaOWaam WaaeaacaaIXaGaey4kaSIaaGOmamaabmaabaWaaSGbaeaacqaHhpWy daWgaaWcbaGaaGOmaaqabaaakeaacqaHhpWydaWgaaWcbaGaaGymaa qabaaaaaGccaGLOaGaayzkaaGaey4kaSYaaeWaaeaadaWcgaqaaiab eE8aJnaaBaaaleaacaaIYaaabeaaaOqaaiabeE8aJnaaBaaaleaaca aIXaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaa kiaawUfacaGLDbaaaiaawIcacaGLPaaaaaa@52B9@ and the amplitude of the new soliton and decrease its width due to an increase of [24-26] when compared with Davydov soliton [3], where μ D = x 1 2 / w(1 s 2 )J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadseaaeqaaOGaeyypa0ZaaSGbaeaacaWG4bWaa0baaSqa aiaaigdaaeaacaaIYaaaaaGcbaGaam4DaiaacIcacaaIXaGaeyOeI0 Iaam4CamaaCaaaleqabaGaaGOmaaaakiaacMcacaWGkbaaaaaa@4328@ , and G D = 4 x 1 2 / w(1 s 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGebaabeaakiabg2da9maalyaabaGaaGinaiaadIhadaqh aaWcbaGaaGymaaqaaiaaikdaaaaakeaacaWG3bGaaiikaiaaigdacq GHsislcaWGZbWaaWbaaSqabeaacaaIYaaaaOGaaiykaaaaaaa@422D@ are the corresponding values in the Davydov mode [3-8]. Thus the localized feature of the new soliton is enhanced. Therefore its stability against the quantum fluctuation and thermal perturbations increased considerably as compared with the Davydov soliton.
V. The properties of carrier of Bio-energy transport
The energy of soliton in Pang’s model becomes [24-26] E=<Φ(t)|H|Φ(t)= 1 r 0 2 [ J r 0 2 ( ϕ x ) 2 +R | ϕ(x,t) | 2 G p | ϕ(x,t) | 4 dx+ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9iabgYda8iabfA6agjaacIcacaWG0bGaaiykaiaacYhacaWGibGa aiiFaiabfA6agjaacIcacaWG0bGaaiykaiabg2da9maalaaabaGaaG ymaaqaaiaadkhadaWgaaWcbaGaaGimaaqabaaaaOWaa8qmaeaacaaI YaWaamqaaeaacaWGkbGaamOCamaaDaaaleaacaaIWaaabaGaaGOmaa aaaOGaay5waaaaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4k IipakmaabmaabaWaaSaaaeaacqGHciITcqaHvpGzaeaacqGHciITca WG4baaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUca Riaadkfadaabdaqaaiabew9aMjaacIcacaWG4bGaaiilaiaadshaca GGPaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia am4ramaaBaaaleaacaWGWbaabeaakmaaemaabaGaeqy1dyMaaiikai aadIhacaGGSaGaamiDaiaacMcaaiaawEa7caGLiWoadaahaaWcbeqa aiaaisdaaaGccaWGKbGaamiEaiabgUcaRaaa@754E@ 1 r 0 1 2 [ M ( β(x,t) t ) 2 +w r 0 ( β(x,t) x ) 2 ] dx= E 0 + 1 2 M sol v 2    (31) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamOCamaaBaaaleaacaaIWaaabeaaaaGcdaWdXaqaamaa laaabaGaaGymaaqaaiaaikdaaaWaamWaaeaacaWGnbWaaeWaaeaada WcaaqaaiabgkGi2kabek7aIjaacIcacaWG4bGaaiilaiaadshacaGG PaaabaGaeyOaIyRaamiDaaaaaiaawIcacaGLPaaadaahaaWcbeqaai aaikdaaaGccqGHRaWkcaWG3bGaamOCamaaDaaaleaacaaIWaaabaaa aOWaaeWaaeaadaWcaaqaaiabgkGi2kabek7aIjaacIcacaWG4bGaai ilaiaadshacaGGPaaabaGaeyOaIyRaamiEaaaaaiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaaaSqaaiabgkHiTi abg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaamizaiaadIhacqGH9aqp caWGfbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaaSaaaeaacaaIXa aabaGaaGOmaaaacaWGnbWaaSbaaSqaaiaadohacaWGVbGaamiBaaqa baGccaWG2bWaaWbaaSqabeaacaaIYaaaaOGaaeiiaiaabccacaqGGa GaaeikaiaabodacaqGXaGaaeykaaaa@702C@ The rest energy of the new soliton is E 0 =2( ε 0 2J) 8 ( x 1 + x 2 ) 4 3 w 2 J = E s 0 W   (32) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIWaaabeaakiabg2da9iaaikdacaGGOaGaeqyTdu2aaSba aSqaaiaaicdaaeqaaOGaeyOeI0IaaGOmaiaadQeacaGGPaGaeyOeI0 YaaSaaaeaacaaI4aGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGc cqGHRaWkcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykamaaCaaale qabaGaaGinaaaaaOqaaiaaiodacaWG3bWaaWbaaSqabeaacaaIYaaa aOGaamOsaaaacqGH9aqpcaWGfbWaa0baaSqaaiaadohaaeaacaaIWa aaaOGaeyOeI0Iaam4vaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGa aeOmaiaabMcaaaa@564D@ where W= [2 ( x 1 + x 2 ) 4 ] / 3 w 2 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maalyaabaGaai4waiaaikdacaGGOaGaamiEamaaBaaaleaacaaI XaaabeaakiabgUcaRiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPa WaaWbaaSqabeaacaaI0aaaaOGaaiyxaaqaaiaaiodacaWG3bWaaWba aSqabeaacaaIYaaaaOGaamOsaaaaaaa@44F1@ is the energy of deformation of the lattice. The effective mass of the new soliton is M sol =2 m ex + 8 ( x 1 + x 2 ) 4 ( 9 s 2 +23 s 4 ) 3 w 2 J ( 1 s 2 ) 3 v 0 2    (33) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGZbGaam4BaiaadYgaaeqaaOGaeyypa0JaaGOmaiaad2ga daWgaaWcbaGaamyzaiaadIhaaeqaaOGaey4kaSYaaSaaaeaacaaI4a WaaeWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamiE amaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaamaaCaaaleqaba GaaGinaaaakmaabmaabaGaaGyoaiaadohadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaaIYaGaeyOeI0IaaG4maiaadohadaahaaWcbeqaai aaisdaaaaakiaawIcacaGLPaaaaeaacaaIZaGaam4DamaaCaaaleqa baGaaGOmaaaakiaadQeadaqadaqaaiaaigdacqGHsislcaWGZbWaaW baaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI ZaaaaOGaamODamaaDaaaleaacaaIWaaabaGaaGOmaaaaaaGccaqGGa GaaeiiaiaabccacaqGOaGaae4maiaabodacaqGPaaaaa@61EA@ We utilize Eqs.(8) and (12)-(14) in the above calculations. In such a case, the binding energy of the new soliton is E BP = 8 ( x 1 + x 2 ) 4 3J w 2     (34) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiuaaqabaGccqGH9aqpdaWcaaqaaiabgkHiTiaa iIdadaqadaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkca WG4bWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaI0aaaaaGcbaGaaG4maiaadQeacaWG3bWaaWbaaSqabeaaca aIYaaaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae4maiaa bsdacaqGPaaaaa@4B61@ E BP MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiuaaqabaaaaa@388A@ is larger than that of the Davydov soliton. The latter is E BD = x 1 4 / 3J w 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiraaqabaGccqGH9aqpcqGHsisldaWcgaqaaiaa dIhadaqhaaWcbaGaaGymaaqaaiaaisdaaaaakeaacaaIZaGaamOsai aadEhadaahaaWcbeqaaiaaikdaaaaaaaaa@40AF@ .They have the following relation: E BP =8 E BD [ 1+4( x 2 x 1 )+6 ( x 2 x 1 ) 2 +4 ( x 2 x 1 ) 3 + ( x 2 x 1 ) 4 ]    (35) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiuaaqabaGccqGH9aqpcaaI4aGaamyramaaBaaa leaacaWGcbGaamiraaqabaGcdaWadaqaaiaaigdacqGHRaWkcaaI0a WaaeWaaeaadaWcaaqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaakeaa caWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaiabgU caRiaaiAdadaqadaqaamaalaaabaGaamiEamaaBaaaleaacaaIYaaa beaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinamaabmaabaWa aSaaaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa iodaaaGccqGHRaWkdaqadaqaamaalaaabaGaamiEamaaBaaaleaaca aIYaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaI0aaaaaGccaGLBbGaayzxaaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGaaeynaiaabMcaaaa@6362@ We can estimate that the binding energy of the new soliton is about several decades larger than that of the Davydov soliton .This is a very interesting result. It is helpful to enhance thermal stability of the new soliton. Obviously, the increase of the binding energy of the new soliton comes from its two-quanta nature and the added interaction i χ 2 ( u n+1 u n )( B n+1 + B n + B n + B n+1 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aHhpWydaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiaadwhadaWgaaWc baGaamOBaiabgUcaRiaaigdaaeqaaOGaeyOeI0IaamyDamaaBaaale aacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamOqamaaDaaa leaacaWGUbGaey4kaSIaaGymaaqaaiabgUcaRaaakiaadkeadaWgaa WcbaGaamOBaaqabaGccqGHRaWkcaWGcbWaa0baaSqaaiaad6gaaeaa cqGHRaWkaaGccaWGcbWaa0baaSqaaiaad6gacqGHRaWkcaaIXaaaba aaaaGccaGLOaGaayzkaaaaleaacaWGPbaabeqdcqGHris5aaaa@52F7@ in the Hamiltonian of the systems, Eq.(10). However, we see from Eq.(35) that the former plays the main role in the increase of the binding energy and the enhancement of thermal stability for the new soliton relative to the latter due to χ 2 < χ 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaikdaaeqaaOGaeyipaWJaeq4Xdm2aaSbaaSqaaiaaigda aeqaaaaa@3C43@ . The increase of the binding energy results in significant changes of properties of the new soliton, which are discussed as follows.

In comparing various correlations to this model, it is helpful to consider them as a function of a composite coupling parameter like that of Pouthier and Spatchek et al. [32-35] and Scott [6] again, it is convenient to define another composite parameter [3,24-26] that can be written as 4π α P = ( χ 1 + χ 2 ) 2 / 2w ω D     (36) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGqbaabeaakiabg2da9maalyaabaWa aeWaaeaacqaHhpWydaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHhp WydaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaaakeaacaaIYaGaam4Daiabl+qiOjabeM8a3naaBaaale aacaWGebaabeaaaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaa bodacaqG2aGaaeykaaaa@4FE7@ where ω D = ( w/M ) 1/2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadseaaeqaaOGaeyypa0ZaaeWaaeaadaWcgaqaaiaadEha aeaacaWGnbaaaaGaayjkaiaawMcaamaaCaaaleqabaWaaSGbaeaaca aIXaaabaGaaGOmaaaaaaaaaa@3EF1@ is the band edge for acoustic phonons (Debye frequency). If, 4π α P >1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGqbaabeaakiabg6da+iaaigdaaaa@3CDD@ it is said to be weak. Using widely accepted values for the physical parameters for the alpha helix protein molecule [2-23], J=1.55× 10 22 J.w=(1319.5)N/m .M=(1.171.91)× 10 25 kg MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaigdacaGGUaGaaGynaiaaiwdacqGHxdaTcaaIXaGaaGimamaa CaaaleqabaGaeyOeI0IaaGOmaiaaikdaaaGccaWGkbGaaiOlaiaadE hacqGH9aqpcaGGOaGaaGymaiaaiodacqGHsislcaaIXaGaaGyoaiaa c6cacaaI1aGaaiykamaalyaabaGaamOtaaqaaiaad2gaaaGaaiOlai aad2eacqGH9aqpcaGGOaGaaGymaiaac6cacaaIXaGaaG4naiabgkHi TiaaigdacaGGUaGaaGyoaiaaigdacaGGPaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaikdacaaI1aaaaOGaam4AaiaadEga aaa@5F96@ χ 1 =62× 10 12 N. χ 2 =(1018)× 10 12 N. r 0 =4.5× 10 10 m.   (37) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOnaiaaikdacqGHxdaTcaaI XaGaaGimamaaCaaaleqabaGaeyOeI0IaaGymaiaaikdaaaGccaWGob GaaiOlaiabeE8aJnaaBaaaleaacaaIYaaabeaakiabg2da9iaacIca caaIXaGaaGimaiabgkHiTiaaigdacaaI4aGaaiykaiabgEna0kaaig dacaaIWaWaaWbaaSqabeaacqGHsislcaaIXaGaaGOmaaaakiaad6ea caGGUaGaamOCamaaBaaaleaacaaIWaaabeaakiabg2da9iaaisdaca GGUaGaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacqGHsisl caaIXaGaaGimaaaakiaad2gacaGGUaGaaeiiaiaabccacaqGGaGaae ikaiaabodacaqG3aGaaeykaaaa@64F6@ We can estimate that the coupled constant lies in the region of 4π α P =0.110.273 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGqbaabeaakiabg2da9iaaicdacaGG UaGaaGymaiaaigdacqGHsislcaaIWaGaaiOlaiaaikdacaaI3aGaaG 4maaaa@4398@ , which is not a weakly coupled theory, the coupling strength is enhanced as compared with the Davydov model, the latter is 4π α D =0.0360.045 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGebaabeaakiabg2da9iaaicdacaGG UaGaaGimaiaaiodacaaI2aGaeyOeI0IaaGimaiaac6cacaaIWaGaaG inaiaaiwdaaaa@444A@ Using the notation of Bullough et al. [29,30], Teki et al. [31,32], and Pouthier et al. [33- 35] γ=J/ 2 w D    (38) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSGbaeaacaWGkbaabaGaaGOmaiabl+qiOjaadEhadaWgaaWc baGaamiraaqabaaaaOGaaeiiaiaabccacaqGGaGaaeikaiaabodaca qG4aGaaeykaaaa@421B@ In terms of the two composite parameters, 4π α P MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGqbaabeaaaaa@3B13@ and γ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379F@ ,the soliton binding energy for Pang’s model can be written by E BP /J = 8 ( 4π α P /γ ) 2 /3 , M sol =2 m ex [ 1+ 32 ( 4π α P ) 2 /3 ]   (39) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGfbWaaSbaaSqaaiaadkeacaWGqbaabeaaaOqaaiaadQeaaaGaeyyp a0ZaaSGbaeaacaaI4aWaaeWaaeaadaWcgaqaaiaaisdacqaHapaCcq aHXoqydaWgaaWcbaGaamiuaaqabaaakeaacqaHZoWzaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG4maaaacaGGSaGaam ytamaaBaaaleaacaWGZbGaam4BaiaadYgaaeqaaOGaeyypa0JaaGOm aiaad2gadaWgaaWcbaGaamyzaiaadIhaaeqaaOWaamWaaeaacaaIXa Gaey4kaSYaaSGbaeaacaaIZaGaaGOmamaabmaabaGaaGinaiabec8a Wjabeg7aHnaaBaaaleaacaWGqbaabeaaaOGaayjkaiaawMcaamaaCa aaleqabaGaaGOmaaaaaOqaaiaaiodaaaaacaGLBbGaayzxaaGaaeii aiaabccacaqGGaGaaeikaiaabodacaqG5aGaaeykaaaa@6106@ From the above parameter values, we find γ=0.08 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0JaaGimaiaac6cacaaIWaGaaGioaaaa@3B8D@ Utilizing this value, the E BP /J MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGfbWaaSbaaSqaaiaadkeacaWGqbaabeaaaOqaaiaadQeaaaaaaa@3979@ versus 4πα MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHbaa@3A12@ relations in Eq.(39) are plotted in Fig.1. However, E BP /J = ( 4π α P /γ ) 2 /3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGfbWaaSbaaSqaaiaadkeacaWGqbaabeaaaOqaaiaadQeaaaGaeyyp a0ZaaSGbaeaadaqadaqaamaalyaabaGaaGinaiabec8aWjabeg7aHn aaBaaaleaacaWGqbaabeaaaOqaaiabeo7aNbaaaiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaakeaacaaIZaaaaaaa@44B0@ for the Davydov model (here M sol ' = m ex [ 1+2 ( 4π α P /γ ) 2 /3 ],4π α D = χ 1 2 / 2w ω D ), MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaamytamaaDa aaleaacaWGZbGaam4BaiaadYgaaeaacaGGNaaaaOGaeyypa0JaamyB amaaBaaaleaacaWGLbGaamiEaaqabaGcdaWadaqaaiaaigdacqGHRa WkcaaIYaWaaSGbaeaadaqadaqaamaalyaabaGaaGinaiabec8aWjab eg7aHnaaBaaaleaacaWGqbaabeaaaOqaaiabeo7aNbaaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaaakeaacaaIZaaaaaGaay5waiaa w2faaiaacYcacaaI0aGaeqiWdaNaeqySde2aaSbaaSqaaiaadseaae qaaOGaeyypa0ZaaSGbaeaacqaHhpWydaqhaaWcbaGaaGymaaqaaiaa ikdaaaaakeaacaaIYaGaam4Daiabl+qiOjabeM8a3naaBaaaleaaca WGebaabeaaaaGccaGGPaGaaiilaaaa@5E6E@ then the E BD /J MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGfbWaaSbaaSqaaiaadkeacaWGebaabeaaaOqaaiaadQeaaaaaaa@396D@ versus 4π α D MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHnaaBaaaleaacaWGebaabeaaaaa@3B07@ relation is also plotted in figure 2 from this figure we see that the difference of soliton binding energies between two models becomes larger with increasing 4πα MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHbaa@3A12@ [24-26].
Figure 2: Binding energy ( E B ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw eadaWgaaWcbaGaamOqaaqabaGccaGGPaaaaa@3915@ of the solitons in our model and the Davydov model in units of dipole-dipole interaction energy ( J ) vs The coupled constant, 4πα MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabec 8aWjabeg7aHbaa@3A0F@ , relationship
Also, we see clearly from Eqs. (28)-(32) and (35) that the localized feature of the new soliton is enhanced due to increases of the nonlinear interaction and of the binding energy of the new soliton resulting from the increases of exciton-phonon interaction in Pang’s model. Thus, the stability of the soliton against quantum and thermal fluctuations is also enhanced considerately [24-26].

As a matter of fact, the nonlinear interaction energy forming the new soliton in Pang’s model is G P = 8 ( χ 1 + χ 2 ) 2 / ( 1 s 2 )w =3.8× 10 32 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4ramaaBa aaleaacaqGqbaabeaakiabg2da9maalyaabaGaaeioamaabmaabaGa eq4Xdm2aaSbaaSqaaiaabgdaaeqaaOGaey4kaSIaeq4Xdm2aaSbaaS qaaiaabkdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaqGYaaa aaGcbaWaaeWaaeaacaqGXaGaeyOeI0Iaae4CamaaCaaaleqabaGaae OmaaaaaOGaayjkaiaawMcaaiaadEhaaaGaeyypa0JaaG4maiaac6ca caaI4aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiabgkHiTiaaio dacaaIYaaaaOGaamOsaaaa@5270@ ,and it is larger than the linear dispersion energy J=1.55× 10 32 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOsaiabg2 da9iaabgdacaGGUaGaaeynaiaabwdacqGHxdaTcaqGXaGaaeimamaa CaaaleqabaGaeyOeI0Iaae4maiaabkdaaaGccaqGkbaaaa@4178@ ,i.e., the nonlinear interaction in Pang’s model is so large that it can actually cancel or suppress the linear dispersion effect in the equation of motion ,thus the new soliton is stable in such a case according the soliton theory [2,33-35]. On the other hand, the nonlinear interaction energy in the Davydov model is only G D = 4 χ 1 2 / ( 1 S 2 )w =1.8× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4ramaaBa aaleaacaqGebaabeaakiabg2da9maalyaabaGaaeinaiabeE8aJnaa BaaaleaacaqGXaaabeaakmaaCaaaleqabaGaaeOmaaaaaOqaamaabm aabaGaaeymaiabgkHiTiaabofadaahaaWcbeqaaiaabkdaaaaakiaa wIcacaGLPaaacaWG3baaaiabg2da9iaaigdacaGGUaGaaGioaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacqGHsislcaaIYaGaaGymaaaa kiaadQeaaaa@4D2F@ and it is about three to four times smaller than G P MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4ramaaBa aaleaacaqGqbaabeaaaaa@37BE@ .Therefore, the stability of the Davydov soliton is weaker as compared with the new soliton. Moreover, the binding energy of the new soliton in Pang’s model is E BP =( 4.164.3 )× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGcbGaamiuaaqabaGccqGH9aqpdaqadaqaaiaaisdacaGG UaGaaGymaiaaiAdacqGHsislcaaI0aGaaiOlaiaaiodaaiaawIcaca GLPaaacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaeyOeI0IaaGOm aiaaigdaaaGccaWGkbaaaa@481B@ in Eq.(31), which is somewhat larger than the thermal perturbation energy, k B T=4.13× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWGcbaabeaakiaadsfacqGH9aqpcaaI0aGaaiOlaiaaigda caaIZaGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiabgkHiTiaaik dacaaIXaaaaOGaamOsaaaa@439F@ at 300K MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacaaIWaGaam4saaaa@38F6@ and about four times larger than the Debye energy kΘ= ω D =1.2× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabfI 5arjabg2da9iabl+qiOjabeM8a3naaBaaaleaacaWGebaabeaakiab g2da9iaaigdacaGGUaGaaGOmaiabgEna0kaaigdacaaIWaWaaWbaaS qabeaacqGHsislcaaIYaGaaGymaaaakiaadQeaaaa@477C@ (there ω D MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadseaaeqaaaaa@38B7@ is the Debye frequency). This shows that transition of the new soliton to a delocalized state can be suppressed by the large energy difference between the initial (solitonic) state and final (delocalized) state, which is very difficult to compensate with the energy of the absorbed phonon. Thus ,the new soliton is robust against quantum fluctuations and thermal perturbation, therefore it has a large lifetime and good thermal stability in the region of biological temperature .In practice, according to Schweitzer et al. s studies (i.e the lifetime of the soliton increases as μ p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CC@ and T 0 = V 0 μ p / K B π MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIWaaabeaakiabg2da9maalyaabaGaeS4dHGMaamOvamaa BaaaleaacaaIWaaabeaakiabeY7aTnaaBaaaleaacaWGWbaabeaaaO qaaiaadUeadaWgaaWcbaGaamOqaaqabaGccqaHapaCaaaaaa@4239@ increase at a given temperature)[15] and the above obtained results, an inference could roughly be drawn that the lifetime of the new soliton will increase considerably as compared with that of the Davydov soliton due to the increase of μ p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CC@ and T 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIWaaabeaaaaa@37B4@ because the latter are about three times larger than that of the Davydov model. On the other hand, the binding energy of the Davydov soliton E BD = χ 1 4 / 3w 2 J=0.188× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyramaaBa aaleaacaqGcbGaaeiraaqabaGccqGH9aqpdaWcgaqaaiabeE8aJnaa DaaaleaadaWgaaadbaGaaeymaaqabaaaleaacaqG0aaaaaGcbaGaae 4maiaabEhadaahaaWcbeqaaiaabkdaaaaaaOGaaeOsaiabg2da9iaa icdacaGGUaGaaGymaiaaiIdacaaI4aGaey41aqRaaGymaiaaicdada ahaaWcbeqaaiabgkHiTiaaikdacaaIXaaaaOGaamOsaaaa@4C3C@ and it is about 23 times smaller than that of the new soliton, about 22 times smaller than K B T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGcbaabeaakiaadsfaaaa@389B@ , and about 6 times smaller than K B Θ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGcbaabeaakiabfI5arbaa@3939@ , respectively. Therefore, the Davydov soliton is easily destructed by the thermal perturbation energy and quantum transition effects. Thus it indicates that the Davdov soliton has a very small lifetime, and it is unstable at the biological temperature 300K MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacaaIWaGaam4saaaa@38F6@ .This conclusion is consistent at a qualitative level with the result s of Wang et al [13,14] and Cottingham et al.[15,21].

One can sum up the differences between Pang’s model and Davydov’s model, Eqs.(1)-(4), as follows. First, the parameter μp is increased (μp =2μD [ 1+2( χ 2 χ 1 )+ ( χ 2 χ 1 ) 2 ],). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca aIXaGaey4kaSIaaGOmaiaacIcadaWcaaqaaiabeE8aJnaaBaaaleaa caaIYaaabeaaaOqaaiabeE8aJnaaBaaaleaacaaIXaaabeaaaaGcca GGPaGaey4kaSIaaiikamaalaaabaGaeq4Xdm2aaSbaaSqaaiaaikda aeqaaaGcbaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaaaakiaacMcada ahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaacaGGSaGaaiykaiaa c6caaaa@4B98@ Secondly the nonlinear coupling energy becomes Gp = 8( χ 1 + χ 2 ) w(1 s 2 ) 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aGaaiikaiabeE8aJnaaBaaaleaacaaIXaaabeaakiabgUcaRiab eE8aJnaaBaaaleaacaaIYaaabeaakiaacMcaaeaacaWG3bGaaiikai aaigdacqGHsislcaWGZbWaaWbaaSqabeaacaaIYaaaaOGaaiykaaaa daahaaWcbeqaaiaaikdaaaaaaa@4524@ (Gp=2GD [ 1+2( χ 2 χ 1 )+ ( χ 2 χ 1 ) 2 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca aIXaGaey4kaSIaaGOmaiaacIcadaWcaaqaaiabeE8aJnaaBaaaleaa caaIYaaabeaaaOqaaiabeE8aJnaaBaaaleaacaaIXaaabeaaaaGcca GGPaGaey4kaSIaaiikamaalaaabaGaeq4Xdm2aaSbaaSqaaiaaikda aeqaaaGcbaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaaaakiaacMcada ahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaaaaa@4989@ , where G D = 4 χ 1 2 w(1 s 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGebaabeaakiabg2da9maalaaabaGaaGinaiabeE8aJnaa DaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadEhacaGGOaGaaGymai abgkHiTiaadohadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaaaa@42E1@ is the nonlinear interaction in the Davydov model, resulting from the two-quanta nature and the enhancement of the coupling the coefficient ( χ 1 + χ 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeE 8aJnaaBaaaleaacaaIXaaabeaakiabgUcaRiabeE8aJnaaBaaaleaa caaIYaaabeaakiaacMcaaaa@3D81@ . For α MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3794@ helical protein molecules, and using the parameter values listed in Eq. (37) the values of the main parameters in Pang’s model can be calculated. These values and the corresponding values in the Davydov model are simultaneously listed in table 1.
Table 1: Comparison of parameters used in the Davydov model and our new model

Parameters
Models

μ

G
(×10-21J)

Amplitude of soliton
A’

Width of soliton Δ X(×10-10m)

Binding energy of soliton EB
(×10-21J)

Our
Model

5.94

3.8

1.72

4.95

-7.8

Davydov
model

1.90

1.18

0.974

14.88

-0.188

From table 1 we can see clearly that the new model produces considerable changes in the properties of the new soliton, such as large increase of the nonlinear interaction, binding energy and amplitude of the soliton, and decrease of its width as compared to that of the Davydov soliton. This indicates that the soliton in Pang’s model is more localized and more robust against quantum and the thermal stability has been enhanced [2,27,28] which implies an increase in lifetime for the new soliton. From Eq. (19) it can also be found that the effect of the two-quanta nature is larger than that of the added interaction. We can therefore refer to the new soliton as quasi-coherent.

In the above studies, the influences of quantum and thermal effects on soliton state, which are expected to cause the soliton to decay into delocalized states, we postulate that the model Hamiltonian and the wave function in Pang’s model together give a complete and realistic picture of the interaction properties and allowed states of the protein molecules. The additional interaction term in the Hamiltonian gives better symmetry of interactions. The new wave function is a reasonable choice for the protein molecules because it not only can exhibit the coherent features of collective excitations arising from the nonlinear interaction between the excitons and phonons, but also retain the conservation of number of particles and fulfill the fact that the energy released by the ATP hydrolysis can only excite two quanta. In such a case, using a standard calculating method [2,26]and widely accepted parameters we can calculate the region encompassed of the excitation or the linear extent of the new soliton, ΔX=2π r 0 / μ p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iwaiabg2da9iaaikdacqaHapaCcaWGYbWaaSbaaSqaaiaaicdaaeqa aOGaai4laiabeY7aTnaaBaaaleaacaWGWbaabeaaaaa@4128@ to be greater than the lattice constant r0, i.e., ΔX> r 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iwaiabg6da+iaadkhadaWgaaWcbaGaaGimaaqabaaaaa@3B1D@ as shown in table 1. Conversely, we can explicitly calculate the amplitude squared of the new soliton using Eq. (29) in its rest frame as |φ(x) | 2 = μ p 2 sec h 2 ( μ p x r 0 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiabeA 8aQjaacIcacaWG4bGaaiykaiaacYhadaahaaWcbeqaaiaaikdaaaGc cqGH9aqpdaWcaaqaaiabeY7aTnaaBaaaleaacaWGWbaabeaaaOqaai aaikdaaaGaci4CaiaacwgacaGGJbGaamiAamaaCaaaleqabaGaaGOm aaaakiaacIcadaWcaaqaaiabeY7aTnaaBaaaleaacaWGWbaabeaaki aadIhaaeaacaWGYbWaaSbaaSqaaiaaicdaaeqaaaaakiaacMcaaaa@4D85@ Thus the probability to find the new soliton outside a range of width r0 is about 0.10. This number can be compatible with the continuous approximation since the quasi-coherent soliton can spread over more than one lattice spacing in the system in such a case. This proves that assuming the continuous approximation used in the calculation is valid. Therefore we should believe that the above calculated results obtained from Pang’s model is correct.
The life time of the carrier of Bio-energy transport at biological temperature
Partially Diagonalized form of the Model Hamiltonian
The lifetime of the soliton in the protein molecules is an centre problem in the process of bioenergy transport because the soliton possess certain biological meanings and can play an important role in the biological process, only if it has enough long lifetimes. Therefore, to calculate the lifetime of the new soliton in Pang’s model has important significance.

For convenlence of calculation, we here represent the wave function of the system in Eq. (9) by [24-26] |Φ(t)>=|ϕ(t)>|β(t)>= U 1 10 ex U 2 10 ph ,     (40) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiabfA 6agjaacIcacaWG0bGaaiykaiabg6da+iabg2da9iaacYhacqaHvpGz caGGOaGaamiDaiaacMcacqGH+aGpcaGG8bGaeqOSdiMaaiikaiaads hacaGGPaGaeyOpa4Jaeyypa0deaaaaaaaaa8qacaqGvbWdamaaBaaa leaapeGaaeymaaWdaeqaaOWdbiaabgdacaqGWaWexLMBbXgBd9gzLb vyNv2CaeHbbr2BIvgigfMBNn3BUDgitrhzGifaiuaacaWFE8=damaa BaaaleaapeGaaeyzaiaabIhaa8aabeaak8qacaqGvbWdamaaBaaale aapeGaaeOmaaWdaeqaaOWdbiaabgdacaqGWaGaa8Nh==aadaWgaaWc baWdbiaabchacaqGObaapaqabaGcpeGaaiilaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeikaiaabsdacaqGWaGaaeykaaaa@6C3A@ where U 1 = 1 λ [1+ n ϕ n (t) B n + + 1 2! ( n ϕ n (t) B n + ) 2 ]     (40a) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGvbWdamaaBaaaleaapeGaaeymaaWdaeqaaOWdbiabg2da98aa daWcaaqaaiaaigdaaeaacqaH7oaBaaGaai4waiaaigdacqGHRaWkda aeqbqaaiabew9aMnaaBaaaleaacaWGUbaabeaakiaacIcacaWG0bGa aiykaiaadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiabgUcaRa WcbaGaamOBaaqab0GaeyyeIuoakmaalaaabaGaaGymaaqaaiaaikda caGGHaaaaiaacIcadaaeqbqaaiabew9aMnaaBaaaleaacaWGUbaabe aakiaacIcacaWG0bGaaiykaiaadkeadaqhaaWcbaGaamOBaaqaaiab gUcaRaaakiaacMcadaahaaWcbeqaaiaaikdaaaaabaGaamOBaaqab0 GaeyyeIuoakiaac2facaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bIcacaqG0aGaaeimaiaabggacaqGPaaaaa@618B@ U 2 =exp{ i n [ β n (t) P n π n (t) u n ] }      (40b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGvbWdamaaBaaaleaapeGaaeOmaaWdaeqaaOWdbiabg2da98aa ciGGLbGaaiiEaiaacchadaGadaqaaiabgkHiTmaalaaabaGaamyAaa qaaiabl+qiObaadaaeqbqaaiaacUfacqaHYoGydaWgaaWcbaGaamOB aaqabaGccaGGOaGaamiDaiaacMcacaWGqbWaaSbaaSqaaiaad6gaae qaaOGaeyOeI0IaeqiWda3aaSbaaSqaaiaad6gaaeqaaOGaaiikaiaa dshacaGGPaGaamyDamaaBaaaleaacaWGUbaabeaaaeaacaWGUbaabe qdcqGHris5aOGaaiyxaaGaay5Eaiaaw2haaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqG0aGaaeimaiaabkgacaqGPa aaaa@5CF0@ =exp{ 1 N q α q (t) a q + α q * (t) a q }     (40c) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaci yzaiaacIhacaGGWbWaaiWaaeaadaWcaaqaaiaaigdaaeaadaGcaaqa aiaad6eaaSqabaaaaOWaaabuaeaacqaHXoqydaWgaaWcbaGaamyCaa qabaGccaGGOaGaamiDaiaacMcacaWGHbWaa0baaSqaaiaadghaaeaa cqGHRaWkaaGccqGHsislcqaHXoqydaqhaaWcbaGaamyCaaqaaiaacQ caaaGccaGGOaGaamiDaiaacMcacaWGHbWaaSbaaSqaaiaadghaaeqa aaqaaiaadghaaeqaniabggHiLdaakiaawUhacaGL9baacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqG0aGaaeimaiaabogacaqG Paaaaa@588B@ where we assume i | ϕ i | 2 =n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaaca GG8bGaeqy1dy2aaSbaaSqaaiaadMgaaeqaaOGaaiiFaaWcbaGaamyA aaqab0GaeyyeIuoakmaaCaaaleqabaGaaGOmaaaakiabg2da9iaad6 gaaaa@40E8@ ,where n is an integer, denotes the number of particle. The wave function, Eq.(40), does not only exhibit coherent properties, but also agrees with the fact that the energy released in the ATP hydrolysis (about 0.43eV) excites only two amide-I vibrational quanta, instead of multiquanta (n>2)[24- 26]. Therefore, the Hamitonian and wave function of the systems, Eqs. (9)-(10), or (40) are reasonable and appropriate to the protein molecules. Using the standard transformation in Eq.(16), where ω q =2 (w/M) 1/2 sin( r 0 q 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadghaaeqaaOGaeyypa0dcbaGaa8Nmaiaa=HcacaWG3bGa a83laiaad2eacaWFPaWaaWbaaSqabeaacaWFXaGaa83laiaa=jdaaa GcciGGZbGaaiyAaiaac6gacaWFOaWaaSaaaeaacaWFYbWaaSbaaSqa aiaa=bdaaeqaaOGaamyCaaqaaiaaikdaaaGaa8xkaaaa@4894@ ,Eq.(10) becomes H= n [ ε 0 B n + B n J( B n + B n+1 + B n+1 + B n ) ] + q ω q ( a q + a q + 1 2 ) + 1 N q.n [ g 1 (q) B n + B n + g 2 (q)( B n + B n+1 + B n + B n+1 )]( a q + a q + ) e in r 0 q    (41) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGib Gaeyypa0ZaaabuaeaadaWadaqaaiabew7aLnaaBaaaleaaieaacaWF WaaabeaakiaadkeadaqhaaWcbaGaamOBaaqaaiabgUcaRaaakiaadk eadaqhaaWcbaGaamOBaaqaaaaakiabgkHiTiaadQeacaWFOaGaamOq amaaDaaaleaacaWGUbaabaGaey4kaScaaOGaamOqamaaDaaaleaaca WGUbGaey4kaSIaa8xmaaqaaaaakiabgUcaRiaadkeadaqhaaWcbaGa amOBaiabgUcaRiaa=fdaaeaacqGHRaWkaaGccaWGcbWaa0baaSqaai aad6gaaeaaaaGccaWFPaaacaGLBbGaayzxaaaaleaacaWGUbaabeqd cqGHris5aOGaey4kaSYaaabuaeaacqWIpecAcqaHjpWDdaWgaaWcba GaamyCaaqabaGccaWFOaGaamyyamaaDaaaleaacaWGXbaabaGaey4k aScaaOGaamyyamaaBaaaleaacaWGXbaabeaakiabgUcaRmaalaaaba Gaa8xmaaqaaiaa=jdaaaGaa8xkaaWcbaGaamyCaaqab0GaeyyeIuoa aOqaaiabgUcaRmaalaaabaGaa8xmaaqaamaakaaabaGaamOtaaWcbe aaaaGcdaaeqbqaaiaa=TfacaWGNbWaaSbaaSqaaiaa=fdaaeqaaOGa a8hkaiaadghacaWFPaGaamOqamaaDaaaleaacaWGUbaabaGaey4kaS caaOGaamOqamaaDaaaleaacaWGUbaabaaaaOGaey4kaSIaam4zamaa BaaaleaacaWFYaaabeaakiaa=HcacaWGXbGaa8xkaiaa=HcacaWGcb Waa0baaSqaaiaad6gaaeaacqGHRaWkaaGccaWGcbWaa0baaSqaaiaa d6gacqGHRaWkcaWFXaaabaaaaOGaey4kaSIaamOqamaaDaaaleaaca WGUbaabaGaey4kaScaaOGaamOqamaaDaaaleaacaWGUbGaey4kaSIa a8xmaaqaaaaakiaa=LcacaWFDbGaa8hkaiaadggadaWgaaWcbaGaam yCaaqabaGccqGHRaWkcaWGHbWaa0baaSqaaiabgkHiTiaadghaaeaa cqGHRaWkaaGccaWFPaGaa8xzamaaCaaaleqabaGaamyAaiaad6gaca WGYbWaa0baaWqaaiaa=bdaaeaaaaWccaWGXbaaaaqaaiaadghacaWF UaGaa8NBaaqab0GaeyyeIuoakiaabccacaqGGaGaaeiiaiaabIcaca qG0aGaaeymaiaabMcaaaaa@9E26@ where g 1 (q)=2 χ 1 i [ 2M ω q ] 1/2 sin r 0 q; g 2 (q)= χ 2 [ 2M ω q ] 1/2 ( e i r 0 q 1)       (42) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaaieaacaWFXaaabeaakiaa=HcacaWGXbGaa8xkaiabg2da9iaa =jdacqaHhpWydaWgaaWcbaGaa8xmaaqabaGccaWGPbWaamWaaeaada Wcaaqaaiabl+qiObqaaiaa=jdacaWGnbGaeqyYdC3aaSbaaSqaaiaa dghaaeqaaaaaaOGaay5waiaaw2faamaaCaaaleqabaGaa8xmaiaa=9 cacaWFYaaaaOGaci4CaiaacMgacaGGUbGaamOCamaaBaaaleaacaWF WaaabeaakiaadghacaWF7aGaa8hiaiaa=bcacaWFGaGaa83zamaaBa aaleaacaWFYaaabeaakiaa=HcacaWGXbGaa8xkaiabg2da9iabeE8a JnaaBaaaleaacaWFYaaabeaakmaadmaabaWaaSaaaeaacqWIpecAae aacaWFYaGaamytaiabeM8a3naaBaaaleaacaWGXbaabeaaaaaakiaa wUfacaGLDbaadaahaaWcbeqaaiaa=fdacaWFVaGaa8Nmaaaakiaa=H cacaWGLbWaaWbaaSqabeaacaWGPbGaamOCamaaBaaameaacaaIWaaa beaaliaadghaaaGccqGHsislcaWFXaGaa8xkaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeinaiaabkdacaqG Paaaaa@726E@ In a semi classical and continuum approximations, from Eq.(41) we can obtain the envelope soliton solution Eq.(29) in Pang’s model, we now represent Eq.(29) by the following form[24-26] ϕ(x,t)= ( μ p 2 ) 1/2 Sech[ μ p r 0 (xvt)]exp[ i ( 2 vx 2J r 0 2 E sol t ) ]      (43) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGSaGaamiDaiaacMcacqGH9aqpdaqadaqaamaalaaa baGaeqiVd02aaSbaaSqaaiaadchaaeqaaaGcbaGaaGOmaaaaaiaawI cacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaakiaadofa caWGLbGaam4yaiaadIgacaGGBbWaaSaaaeaacqaH8oqBdaWgaaWcba GaamiCaaqabaaakeaaieaacaWFYbWaaSbaaSqaaiaa=bdaaeqaaaaa kiaacIcacaWG4bGaeyOeI0IaamODaiaadshacaGGPaGaaiyxaiGacw gacaGG4bGaaiiCamaadmaabaWaaSaaaeaacaWGPbaabaGaeS4dHGga amaabmaabaWaaSaaaeaacqWIpecAdaahaaWcbeqaaiaaikdaaaGcca WG2bGaamiEaaqaaiaaikdacaWGkbGaamOCamaaDaaaleaacaaIWaaa baGaaGOmaaaaaaGccqGHsislcaWGfbWaaSbaaSqaaiaadohacaWGVb GaamiBaaqabaGccaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabsdaca qGZaGaaeykaaaa@70E4@ where μ p = 2 ( χ 1 + χ 2 ) 2 w(1 s 2 )J      (44) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaOGaeyypa0ZaaSaaaeaaieaacaWFYaGaa8hk aiabeE8aJnaaBaaaleaacaWFXaaabeaakiabgUcaRiabeE8aJnaaBa aaleaacaWFYaaabeaakiaa=LcadaahaaWcbeqaaiaa=jdaaaaakeaa caWG3bGaa8hkaiaa=fdacqGHsislcaWGZbWaaWbaaSqabeaacaWFYa aaaOGaa8xkaiaa=PeaaaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGOaGaaeinaiaabsdacaqGPaaaaa@4F95@ The energy of the new soliton is α q (t)= iπ( χ 1 + χ 2 ) w μ p (1 v 2 / v 0 2 ) [ M 2 ω q ] 1 2 ( ω q +qv)csch(πq r 0 /2 μ p ) e iqvt = α q e iqvt     (46) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadghaaeqaaGqaaOGaa8hkaiaadshacaWFPaGaeyypa0Za aSaaaeaacaWGPbGaeqiWdaNaa8hkaiabeE8aJnaaBaaaleaacaWFXa aabeaakiabgUcaRiabeE8aJnaaBaaaleaacaWFYaaabeaakiaa=Lca aeaacaWG3bGaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaa8hkaiaa=f dacqGHsislcaWG2bWaaWbaaSqabeaacaWFYaaaaOGaa83laiaadAha daqhaaWcbaGaa8hmaaqaaiaa=jdaaaGccaWFPaaaamaadmaabaWaaS aaaeaacaWGnbaabaGaa8Nmaiabl+qiOjabeM8a3naaBaaaleaacaWG XbaabeaaaaaakiaawUfacaGLDbaadaahaaWcbeqaamaaliaabaGaa8 xmaaqaaiaa=jdaaaaaaOGaa8hkaiabeM8a3naaBaaaleaacaWGXbaa beaakiabgUcaRiaadghacaWG2bGaa8xkaiGacogacaGGZbGaai4yai aadIgacaWFOaGaeqiWdaNaamyCaiaadkhadaWgaaWcbaGaa8hmaaqa baGccaWFVaGaa8NmaiabeY7aTnaaBaaaleaacaWGWbaabeaakiaa=L cacaWGLbWaaWbaaSqabeaacaWGPbGaamyCaiaadAhacaWG0baaaOGa eyypa0JaeqySde2aaSbaaSqaaiaadghaaeqaaOGaamyzamaaCaaale qabaGaamyAaiaadghacaWG2bGaamiDaaaakiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeinaiaabAdacaqGPaaaaa@83AC@ This treatment yields a localized coherent structure of the excitons with size of order 2 r0/μp that propagates with velocity v and can transfer energy ES01< . Unlike bare excitons that are scattered by the interactions with the phonons, this soliton state describes a quasi-particle consisting of the two excitons plus a lattice deformation and hence a priori includes interaction with the acoustic phonons. So the soliton is not scattered and spread by this interaction of the vibration of amino acids (lattices), and can maintain its form, energy, momentum and other quasiparticle properties moving over a macroscopic distance. The bell-shaped form of the soliton Eq. (43) does not depend on the excitation method. It is self-consistent. Since the soliton always move with velocity less than that of longitudinal sound in the chain they do not emit phonons, i.e., their kinetic energy is not transformed into thermal energy. This is one important reason for the high stability of the new soliton. In addition the energy of the soliton state is below the bottom of the bare exciton bands, the energy gap being for small velocity of propagation. Hence there is an energy penalty associated with the destruction with transformation from the soliton state to a bare exciton state, i.e, the destruction of the soliton state requires simultaneous removal of the lattice distortion. We know in general that the transition probability to a lattice state without distortion is very small, in general, being negligible for a long chain. Considering this it is reasonable to assume that such a soliton is stable enough to propagate through the length of a typical protein structure. However, the thermal stability of the soliton state must be calculated quantitatively. The following calculation addresses this point explicitly [24- 26].We now diagonalize partially the model Hamiltonian in order to calculate the lifetime of the soliton, Eq. (43), using the quantum perturbation method [14] Since one is interested in investigating the case where there is initially a soliton moving with a velocity v on the chains, it is convenlent to do the analysis in a frame of reference where the soliton is at rest. We should then consider the Hamiltonian in this rest frame of the soliton, H ˜ vP, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aeaaaaaaaaa8qacqGHsislcaqG2bGaaeiuaiaacYcaaaa@3A5A@ where P is the total momentum, and P= q q( a q + a q B q + B q ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaeyypa0ZdamaaqafabaGaeS4dHGMaamyCaGqaaiaa=Hca caWGHbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccaWGHbWaaSbaaS qaaiaadghaaeqaaOGaeyOeI0IaamOqamaaDaaaleaacaWGXbaabaGa ey4kaScaaOGaamOqamaaBaaaleaacaWGXbaabeaakiaa=LcaaSqaai aadghaaeqaniabggHiLdaaaa@494E@ ,where B q + = 1 N n e iqn r 0 B n + MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaDa aaleaacaWGXbaabaGaey4kaScaaOGaeyypa0ZaaSaaaeaacaaIXaaa baWaaOaaaeaacaWGobaaleqaaaaakmaaqafabaGaamyzamaaCaaale qabaGaamyAaiaadghacaWGUbGaamOCamaaBaaameaacaaIWaaabeaa aaaaleaacaWGUbaabeqdcqGHris5aOGaamOqamaaDaaaleaacaWGUb aabaGaey4kaScaaaaa@4749@ . Also, in order to have simple analytical expressions we make the usual continuum approximation. This gives H ˜ = 0 L dx2[ ( ε 0 2J) ϕ + (x)ϕ(x)+J r 0 2 ϕ + x ϕ x iv 2 ( ϕ + x ϕ(x) ϕ + (x) ϕ x ) ] + q ( ω q qv) a q + a q + 1 N q 2[ g 1 (q)+2 g 2 (q)]( a q + + a q ) 0 L dx e ikx ϕ + (x) ϕ (x)    (47) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaaceWGib GbaGaacqGH9aqpdaWdXaqaaiaadsgacaWG4bacbaGaa8Nmamaadmaa baGaa8hkaiabew7aLnaaBaaaleaacaWFWaaabeaakiabgkHiTiaa=j dacaWGkbGaa8xkaiabew9aMnaaCaaaleqabaGaey4kaScaaOGaa8hk aiaadIhacaWFPaGaeqy1dyMaa8hkaiaadIhacaWFPaGaey4kaSIaam OsaiaadkhadaqhaaWcbaGaa8hmaaqaaiaa=jdaaaGcdaWcaaqaaiab gkGi2kabew9aMnaaCaaaleqabaGaey4kaScaaaGcbaGaeyOaIyRaam iEaaaadaWcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadIhaaaGa eyOeI0YaaSaaaeaacaWGPbGaeS4dHGMaamODaaqaaiaa=jdaaaWaae WaaeaadaWcaaqaaiabgkGi2kabew9aMnaaCaaaleqabaGaey4kaSca aaGcbaGaeyOaIyRaamiEaaaacqaHvpGzcaWFOaGaamiEaiaa=Lcacq GHsislcqaHvpGzdaahaaWcbeqaaiabgUcaRaaakiaa=HcacaWG4bGa a8xkamaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaamiEaaaaai aawIcacaGLPaaaaiaawUfacaGLDbaaaSqaaiaa=bdaaeaacaWGmbaa niabgUIiYdaakeaacqGHRaWkdaaeqbqaaiabl+qiOjaa=HcacqaHjp WDdaWgaaWcbaGaamyCaaqabaGccqGHsislcaWGXbGaamODaiaa=Lca caWGHbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccaWGHbWaaSbaaS qaaiaadghaaeqaaOGaey4kaSYaaSaaaeaacaWFXaaabaWaaOaaaeaa caWGobaaleqaaaaakmaaqafabaaaleaacaWGXbaabeqdcqGHris5aO Gaa8Nmaiaa=TfacaWGNbWaaSbaaSqaaiaa=fdaaeqaaOGaa8hkaiaa dghacaWFPaGaey4kaSIaa8NmaiaadEgadaWgaaWcbaGaa8Nmaaqaba GccaWFOaGaamyCaiaa=LcacaWFDbGaa8hkaiaadggadaqhaaWcbaGa eyOeI0IaamyCaaqaaiabgUcaRaaakiabgUcaRiaadggadaWgaaWcba GaamyCaaqabaGccaWFPaWaa8qmaeaacaWGKbGaamiEaiaadwgadaah aaWcbeqaaiaadMgacaWGRbGaamiEaaaakiabew9aMnaaCaaaleqaba Gaey4kaScaaOGaa8hkaiaadIhacaWFPaGaeqy1dy2aaWbaaSqabeaa aaGccaWFOaGaamiEaiaa=LcaaSqaaiaa=bdaaeaacaWGmbaaniabgU IiYdaaleaacaWGXbaabeqdcqGHris5aOGaaeiiaiaabccacaqGGaGa aeikaiaabsdacaqG3aGaaeykaaaaaa@C07F@ where  ϕ(x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiabew 9aMHqaaiaa=HcacaWG4bGaa8xkaaaa@3AB9@ represents now the field operator corresponding to Bn in the continuum limit (whereas before it only indicated a numerical value). Here L=Nr0, -π< kr0< π, and ω q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadghaaeqaaaaa@38E4@ ≈(w/M)1/2 r0•|q| ,x=nr0. Since the soliton excitation is connected with the deformation of intermolecular spacing, it is necessary to pass in Eq.(47) to new phonons taking this deformation into account. Such a transformation can be realized by means of the following transformation of phonon operators [29] b q = a p 1 N α q , b q + = a q + 1 N α q * ,   (48) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGIbWdamaaBaaaleaapeGaaeyCaaWdaeqaaOWdbiabg2da98aa caWGHbWaaSbaaSqaaiaadchaaeqaaOGaeyOeI0YaaSaaaeaaieaaca WFXaaabaWaaOaaaeaacaWGobaaleqaaaaakiabeg7aHnaaBaaaleaa caWGXbaabeaakiaa=XcacaWFGaGaa8hiaiaa=jgadaqhaaWcbaGaa8 xCaaqaaiabgUcaRaaakiabg2da9iaadggadaqhaaWcbaGaamyCaaqa aiabgUcaRaaakiabgkHiTmaalaaabaGaa8xmaaqaamaakaaabaGaam OtaaWcbeaaaaGccqaHXoqydaqhaaWcbaGaamyCaaqaaiaa=PcaaaGc caWFSaGaaeiiaiaabccacaqGGaGaaeikaiaabsdacaqG4aGaaeykaa aa@5558@ which describe phonons relative to a chain with a particular deformation, where bq (bq+) is the annihilation (creation) operator of new phonon. The vacuum state for the new phonons is | 0 ~ ph =exp[ 1 N q ( α q (t) a q + α q * (t)) a q ]|0> ph     (49) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFam aaxacabaGaa8hmaaWcbeGcbaGaa8NFaaaacqGHQms8daWgaaWcbaGa amiCaiaadIgaaeqaaOGaeyypa0JaciyzaiaacIhacaGGWbWaamWaae aadaWcaaqaaiaa=fdaaeaadaGcaaqaaiaad6eaaSqabaaaaOWaaabu aeaacaWFOaGaeqySde2aaSbaaSqaaiaadghaaeqaaOGaa8hkaiaads hacaWFPaGaamyyamaaDaaaleaacaWGXbaabaGaey4kaScaaOGaeyOe I0IaeqySde2aa0baaSqaaiaadghaaeaacaWFQaaaaOGaa8hkaiaa=r hacaWFPaGaa8xkaiaa=fgadaWgaaWcbaGaa8xCaaqabaaabaGaamyC aaqab0GaeyyeIuoaaOGaay5waiaaw2faaiaa=XhacaWFWaGaeyOpa4 Jaa8hiamaaBaaaleaacaWGWbGaamiAaaqabaGccaqGGaGaaeiiaiaa bccacaqGGaGaaeikaiaabsdacaqG5aGaaeykaaaa@6455@ which is a coherent phonon state[30], i.e., | 0 ˜ ph =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFai qa=bdagaacaiabgQYiXpaaBaaaleaacaWGWbGaamiAaaqabaGccqGH 9aqpcaWFWaaaaa@3D54@ . The Hamiltonian H ~ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca WGibaaleqakeaacaGG+baaaaaa@3817@ can now be rewritten[ 24-26] as H ~ = 0 L 2dxϕ(x)[ ε 0 2J+V(x)J r 0 2 2 x 2 +i x ] ϕ(x)+ q ( ω q qv)[ b q + b q + 1 N ( α q b q + + α q * b q + )] +W'+ 1 N 2[ g 1 (q)+2 g 2 (q)]( b q + + b q ) 0 L dx e iqx ϕ + (x) ϕ(x)             (50) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaadaWfGa qaaiaadIeaaSqabOqaaiaac6haaaGaeyypa0Zaa8qmaeaaieaacaWF YaGaamizaiaadIhacqaHvpGzcaWFOaGaamiEaiaa=LcacaWFBbGaeq yTdu2aaSbaaSqaaiaa=bdaaeqaaOGaeyOeI0Iaa8NmaiaadQeacqGH RaWkcaWGwbGaa8hkaiaadIhacaWFPaGaeyOeI0IaamOsaiaadkhada qhaaWcbaGaa8hmaaqaaiaa=jdaaaGcdaWcaaqaaiabgkGi2oaaCaaa leqabaGaa8NmaaaaaOqaaiabgkGi2kaadIhadaahaaWcbeqaaiaa=j daaaaaaOGaey4kaSIaamyAaiabl+qiOnaalaaabaGaeyOaIylabaGa eyOaIyRaamiEaaaacaWFDbaaleaacaWFWaaabaGaamitaaqdcqGHRi I8aOGaeqy1dyMaa8hkaiaadIhacaWFPaGaey4kaScabaGaa8hiaiaa =bcadaaeqbqaaiabl+qiOjaa=HcacqaHjpWDdaWgaaWcbaGaamyCaa qabaGccqGHsislcaWGXbGaamODaiaa=LcacaWFBbGaamOyamaaDaaa leaacaWGXbaabaGaey4kaScaaOGaamOyamaaBaaaleaacaWGXbaabe aakiabgUcaRmaalaaabaGaa8xmaaqaamaakaaabaGaamOtaaWcbeaa aaGccaWFOaGaeqySde2aaSbaaSqaaiaadghaaeqaaOGaamOyamaaDa aaleaacaWGXbaabaGaey4kaScaaOGaey4kaSIaeqySde2aa0baaSqa aiaadghaaeaacaWFQaaaaOGaamOyamaaDaaaleaacaWGXbaabaGaey 4kaScaaOGaa8xkaiaa=1faaSqaaiaa=fhaaeqaniabggHiLdGccqGH RaWkcaWGxbGaai4jaiabgUcaRaqaaiaa=bcacaWFGaWaaSaaaeaaca WFXaaabaWaaOaaaeaacaWGobaaleqaaaaakmaaqaeabaGaa8Nmaiaa =TfacaWGNbWaaSbaaSqaaiaa=fdaaeqaaOGaa8hkaiaadghacaWFPa Gaey4kaSIaa8NmaiaadEgadaWgaaWcbaGaa8NmaaqabaGccaWFOaGa amyCaiaa=LcacaWFDbGaa8hkaiaadkgadaqhaaWcbaGaeyOeI0Iaam yCaaqaaiabgUcaRaaakiabgUcaRiaadkgadaWgaaWcbaGaamyCaaqa baGccaWFPaWaa8qmaeaacaWGKbGaamiEaiaadwgadaahaaWcbeqaai aadMgacaWGXbGaamiEaaaakiabew9aMnaaDaaaleaaaeaacqGHRaWk aaGccaWFOaGaamiEaiaa=LcaaSqaaiaa=bdaaeaacaWGmbaaniabgU IiYdaaleqabeqdcqGHris5aOGaeqy1dyMaa8hkaiaadIhacaWFPaGa a8hiaiaa=bcacaWFGaGaa8hiaiaa=bcacaWFGaGaa8hiaiaa=bcaca WFGaGaa8hiaiaa=bcacaWFGaGaa8hiaiaa=bcacaWFGaGaa8hiaiaa =bcacaWFGaGaa8hiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa=HcacaWF 1aGaa8hmaiaa=Lcaaaaa@CE92@ where W'= 1 N q ( ω q qv)| α q | 2 ,V(x)= 1 N q [ g 1 (q)+2 g 2 (q)]( α q * + α q ) e iqx       (51) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiaacE cacqGH9aqpdaWcaaqaaGqaaiaa=fdaaeaacaWGobaaamaaqafabaGa eS4dHGgaleaacaWGXbaabeqdcqGHris5aOGaa8hkaiabeM8a3naaBa aaleaacaWGXbaabeaakiabgkHiTiaadghacaWG2bGaa8xkaiaa=Xha cqaHXoqydaqhaaWcbaGaamyCaaqaaaaakiaa=XhadaahaaWcbeqaai aa=jdaaaGccaGGSaGaamOvaiaa=HcacaWG4bGaa8xkaiabg2da9maa laaabaGaa8xmaaqaaiaad6eaaaWaaabuaeaacaWFBbGaam4zamaaBa aaleaacaWFXaaabeaakiaa=HcacaWGXbGaa8xkaiabgUcaRiaa=jda caWGNbWaaSbaaSqaaiaa=jdaaeqaaOGaa8hkaiaadghacaWFPaGaa8 xxaiaa=HcacqaHXoqydaqhaaWcbaGaeyOeI0IaamyCaaqaaiaa=Pca aaGccqGHRaWkcqaHXoqydaqhaaWcbaGaeyOeI0IaamyCaaqaaaaaki aa=LcacaWGLbWaaWbaaSqabeaacaWGPbGaamyCaiaadIhaaaaabaGa amyCaaqab0GaeyyeIuoakiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabIcacaqG1aGaaeymaiaabMcaaaa@753D@ To describe the deformation corresponding to a soliton in the subspace where there is 0 L dx ϕ + (x) ϕ (x) =1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGKbGaamiEaiabew9aMnaaDaaaleaaaeaacqGHRaWkaaacbaGccaWF OaGaamiEaiaa=LcacqaHvpGzdaqhaaWcbaaabaaaaOGaa8hkaiaadI hacaWFPaaaleaacaWFWaaabaGaamitaaqdcqGHRiI8aOGaeyypa0Ja aGymaaaa@46DA@ from Eq(45) in such a case. From the above formulae we can obtain V( x )=2J μ p 2 sec h 2 ( μ p x/ r 0 )     (52) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGwbWdamaabmaabaWdbiaabIhaa8aacaGLOaGaayzkaaWdbiab g2da98aacqGHsislcaaIYaGaamOsaiabeY7aTnaaBaaaleaacaWGWb aabeaakmaaCaaaleqabaGaaGOmaaaakiGacohacaGGLbGaai4yaiaa dIgadaahaaWcbeqaaiaaikdaaaGccaGGOaGaeqiVd02aaSbaaSqaai aadchaaeqaaOGaamiEaiaac+cacaWGYbWaaSbaaSqaaiaaicdaaeqa aOGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabw dacaqGYaGaaeykaaaa@537E@ In order to partially diagonalize the Hamiltonian Eq.(50) we introduce the following canonical transformation[14,23] ϕ(x)= j A j C j (x),   ϕ + (x)= j C j * (x) A j +        (53) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGPaGaeyypa0ZaaabuaeaacaWGbbWaaSbaaSqaaiaa dQgaaeqaaaqaaiaadQgaaeqaniabggHiLdGccaWGdbWaaSbaaSqaai aadQgaaeqaaOGaaiikaiaadIhacaGGPaGaaiilaiaabccacaqGGaGa eqy1dy2aaWbaaSqabeaacqGHRaWkaaGccaGGOaGaamiEaiaacMcacq GH9aqpdaaeqbqaaiaadoeadaqhaaWcbaGaamOAaaqaaiaacQcaaaGc caGGOaGaamiEaiaacMcacaWGbbWaa0baaSqaaiaadQgaaeaacqGHRa WkaaaabaGaamOAaaqab0GaeyyeIuoakiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabodacaqGPaaaaa@5E82@ where C 1 * (x) C j (x)dx= δ lj , j C j * ( x ) C j (x)=δ(x x ), dx| C j (x) | 2 =1       (54) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaaie aacaWFdbWaa0baaSqaaiaa=fdaaeaacaWFQaaaaOGaa8hkaiaadIha caWFPaGaam4qamaaBaaaleaacaWGQbaabeaakiaa=HcacaWG4bGaa8 xkaiaadsgacaWG4bGaeyypa0JaeqiTdq2aaSbaaSqaaiaadYgacaWG Qbaabeaakiaa=XcadaaeqbqaaiaadoeadaqhaaWcbaGaamOAaaqaai aa=PcaaaGccaWFOaGabmiEayaafaGaa8xkaiaadoeadaWgaaWcbaGa amOAaaqabaGccaWFOaGaamiEaiaa=LcacqGH9aqpcqaH0oazcaWFOa GaamiEaiabgkHiTiqadIhagaqbaiaa=LcacaWFSaWaa8qaaeaacaWG KbGaamiEaiaacYhacaWGdbWaaSbaaSqaaiaadQgaaeqaaOGaa8hkai aadIhacaWFPaGaa8hFamaaCaaaleqabaGaa8Nmaaaakiabg2da9iaa =fdaaSqabeqaniabgUIiYdaaleaacaWGQbaabeqdcqGHris5aaWcbe qab0Gaey4kIipakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabIcacaqG1aGaaeinaiaabMcaaaa@6F73@ The operators As+ and Ak+ the creation operators for the bound states Cs(x) and delocalized state Ck(x), respectively. The detailed calculation of the partial diagonalization and of corresponding Cs(x) and Ck(x) are described in Appenix A. The partially diagonalized Hamiltonian obtained is as follows H ˜ =W'+ E s A s + A s + k E k A k + A k + q ( ω q qv) b q + b q +    1 N q ( ω q qv)( b q + α q + α q b q )(1 A s + A s ) + 1 N k k q F(k, k ,q)( b q + + b q ) A k + A k    1 N kq F ˜ (k,q)( b q + + b q ) ( A s + A k A k + A s )                                                           (55)       MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaaceWGib GbaGaacqGH9aqpcaWGxbGaai4jaiabgUcaRiaadweadaWgaaWcbaGa am4CaaqabaGccaWGbbWaa0baaSqaaiaadohaaeaacqGHRaWkaaGcca WGbbWaaSbaaSqaaiaadohaaeqaaOGaey4kaSYaaabuaeaacaWGfbWa aSbaaSqaaiaadUgaaeqaaOGaamyqamaaDaaaleaacaWGRbaabaGaey 4kaScaaOGaamyqamaaBaaaleaacaWGRbaabeaakiabgUcaRaWcbaGa am4Aaaqab0GaeyyeIuoakmaaqafabaGaeS4dHGgcbaGaa8hkaiabeM 8a3naaBaaaleaacaWGXbaabeaakiabgkHiTiaadghacaWG2bGaa8xk aiaadkgadaqhaaWcbaGaamyCaaqaaiabgUcaRaaakiaadkgadaWgaa WcbaGaamyCaaqabaaabaGaamyCaaqab0GaeyyeIuoakiabgUcaRaqa aiaabccacaqGGaWaaSaaaeaacaWFXaaabaWaaOaaaeaacaWGobaale qaaaaakmaaqafabaGaeS4dHGMaa8hkaiabeM8a3naaBaaaleaacaWG XbaabeaakiabgkHiTiaadghacaWG2bGaa8xkaiaa=HcacaWGIbWaa0 baaSqaaiaadghaaeaacqGHRaWkaaGccqaHXoqydaWgaaWcbaGaamyC aaqabaGccqGHRaWkcqaHXoqydaqhaaWcbaGaamyCaaqaaiabgEHiQa aakiaadkgadaWgaaWcbaGaamyCaaqabaaabaGaamyCaaqab0Gaeyye Iuoakiaa=LcacaWFOaGaa8xmaiabgkHiTiaadgeadaqhaaWcbaGaam 4CaaqaaiabgUcaRaaakiaadgeadaWgaaWcbaGaam4CaaqabaGccaWF PaGaa8hiaiaa=bcacqGHRaWkdaWcaaqaaiaa=fdaaeaadaGcaaqaai aad6eaaSqabaaaaOWaaabuaeaacaWGgbGaa8hkaiaadUgacaWFSaGa bm4AayaafaGaa8hlaiaadghacaWFPaGaa8hkaiaadkgadaqhaaWcba GaeyOeI0IaamyCaaqaaiabgUcaRaaakiabgUcaRiaadkgadaWgaaWc baGaamyCaaqabaGccaWFPaaaleaacaWGRbGabm4AayaafaGaamyCaa qab0GaeyyeIuoakiaadgeadaqhaaWcbaGabm4AayaafaaabaGaey4k aScaaOGaamyqamaaBaaaleaacaWGRbaabeaaaOqaaiaabccacaqGGa GaeyOeI0YaaSaaaeaacaWFXaaabaWaaOaaaeaacaWGobaaleqaaaaa kmaaqafabaGabmOrayaaiaGaa8hkaiaadUgacaWFSaGaamyCaiaa=L cacaWFOaGaamOyamaaDaaaleaacqGHsislcaWGXbaabaGaey4kaSca aOGaey4kaSIaamOyamaaBaaaleaacaWGXbaabeaakiaa=LcaaSqaai aadUgacaWGXbaabeqdcqGHris5aOGaa8hkaiaadgeadaqhaaWcbaGa am4CaaqaaiabgUcaRaaakiaadgeadaWgaaWcbaGaeyOeI0Iaam4Aaa qabaGccqGHsislcaWGbbWaa0baaSqaaiaadUgaaeaacqGHRaWkaaGc caWGbbWaaSbaaSqaaiaadohaaeqaaOGaa8xkaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaabwdacaqG1aGaaeykaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaaaaaa@EBD2@ and C s (x)= ( μ p 2 r 0 ) 1/2 sech( μ p x/ r 0 )exp[ixv/2J r 0 2 ],    MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbaabeaakiaacIcacaWG4bGaaiykaiabg2da9iaacIca daWcaaqaaiabeY7aTnaaBaaaleaacaWGWbaabeaaaOqaaiaaikdaca WGYbWaaSbaaSqaaiaaicdaaeqaaaaakiaacMcadaahaaWcbeqaaiaa igdacaGGVaGaaGOmaaaakiGacohacaGGLbGaai4yaiaadIgacaGGOa GaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaamiEaiaac+cacaWGYbWa aSbaaSqaaiaaicdaaeqaaOGaaiykaiGacwgacaGG4bGaaiiCaiaacU facaWGPbGaeS4dHGMaamiEaiaadAhacaGGVaGaaGOmaiaadQeacaWG YbWaa0baaSqaaiaaicdaaeaacaaIYaaaaOGaaiyxaiaacYcacaqGGa Gaaeiiaiaabccaaaa@6041@ with  E s =2[ ε 0 2J 2 v 2 2J r 0 2 μ p 2 J ]      (56a) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG3bGaaeyAaiaabshacaqGObGaaeiia8aacaWGfbWaaSbaaSqa aiaadohaaeqaaOGaeyypa0JaaGOmamaadmaabaGaeqyTdu2aaSbaaS qaaiaaicdaaeqaaOGaeyOeI0IaaGOmaiaadQeacqGHsisldaWcaaqa aiabl+qiOnaaCaaaleqabaGaaGOmaaaakiaadAhadaahaaWcbeqaai aaikdaaaaakeaacaaIYaGaamOsaiaadkhadaqhaaWcbaGaaGimaaqa aiaaikdaaaaaaOGaeyOeI0IaeqiVd02aa0baaSqaaiaadchaaeaaca aIYaaaaOGaamOsaaGaay5waiaaw2faaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabIcacaqG1aGaaeOnaiaabggacaqGPaaaaa@5B58@ C k (x)= μ p tanh( μ p x/ r 0 )ik r 0 N r 0 [ μ p ik r 0 ] exp[ikx+ ivx 2J r 0 2 ],        (56b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGRbaabeaakiaacIcacaWG4bGaaiykaiabg2da9maalaaa baGaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaciiDaiaacggacaGGUb GaaiiAaiaacIcacqaH8oqBdaWgaaWcbaGaamiCaaqabaGccaWG4bGa ai4laiaadkhadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyOeI0Iaam yAaiaadUgacaWGYbWaaSbaaSqaaiaaicdaaeqaaaGcbaWaaOaaaeaa caWGobGaamOCamaaBaaaleaacaaIWaaabeaaaeqaaOGaai4waiabeY 7aTnaaBaaaleaacaWGWbaabeaakiabgkHiTiaadMgacaWGRbGaamOC amaaBaaaleaacaaIWaaabeaakiaac2faaaGaciyzaiaacIhacaGGWb Gaai4waiaadMgacaWGRbGaamiEaiabgUcaRmaalaaabaGaamyAaiab l+qiOjaadAhacaWG4baabaGaaGOmaiaadQeacaWGYbWaa0baaSqaai aaicdaaeaacaaIYaaaaaaakiaac2facaGGSaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabA dacaqGIbGaaeykaaaa@74A8@ with E k =2[ ε 0 2J 2 v 2 2J r 0 2 J (k r 0 ) 2 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGRbaabeaakiabg2da9iaaikdadaWadaqaaiabew7aLnaa BaaaleaacaaIWaaabeaakiabgkHiTiaaikdacaWGkbGaeyOeI0YaaS aaaeaacqWIpecAdaahaaWcbeqaaiaaikdaaaGccaWG2bWaaWbaaSqa beaacaaIYaaaaaGcbaGaaGOmaiaadQeacaWGYbWaa0baaSqaaiaaic daaeaacaaIYaaaaaaakiabgkHiTiaadQeacaGGOaGaam4Aaiaadkha daWgaaWcbaGaaGimaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaa GccaGLBbGaayzxaaaaaa@50BD@ where F(k, k ,q)=2[ g 1 (q)+2 g 2 (q)] 0 L dx e iqx C k (x) C k (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacI cacaWGRbGaaiilaiqadUgagaqbaiaacYcacaWGXbGaaiykaiabg2da 9iaaikdacaGGBbGaam4zamaaBaaaleaacaaIXaaabeaakiaacIcaca WGXbGaaiykaiabgUcaRiaaikdacaWGNbWaaSbaaSqaaiaaikdaaeqa aOGaaiikaiaadghacaGGPaGaaiyxamaapedabaGaamizaiaadIhaca WGLbWaaWbaaSqabeaacaWGPbGaamyCaiaadIhaaaGccaWGdbWaa0ba aSqaaiqadUgagaqbaaqaaiabgEHiQaaaaeaacaaIWaaabaGaamitaa qdcqGHRiI8aOGaaiikaiaadIhacaGGPaGaam4qamaaBaaaleaacaWG RbaabeaakiaacIcacaWG4bGaaiykaaaa@5CE9@ 2[ g 1 (q)+2 g 2 (q)]{ 1 i μ p q r 0 [ μ p +i(k+q) r 0 ][ μ p ik r 0 ] }F[k,(k+q),q] δ k k+q      (57) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyisISRaaG OmaiaacUfacaWGNbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadgha caGGPaGaey4kaSIaaGOmaiaadEgadaWgaaWcbaGaaGOmaaqabaGcca GGOaGaamyCaiaacMcacaGGDbWaaiWaaeaacaaIXaGaeyOeI0YaaSaa aeaacaWGPbGaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaamyCaiaadk hadaWgaaWcbaGaaGimaaqabaaakeaacaGGBbGaeqiVd02aaSbaaSqa aiaadchaaeqaaOGaey4kaSIaamyAaiaacIcacaWGRbGaey4kaSIaam yCaiaacMcacaWGYbWaaSbaaSqaaiaaicdaaeqaaOGaaiyxaiaacUfa cqaH8oqBdaWgaaWcbaGaamiCaaqabaGccqGHsislcaWGPbGaam4Aai aadkhadaWgaaWcbaGaaGimaaqabaGccaGGDbaaaaGaay5Eaiaaw2ha aiabgIKi7kaadAeacaGGBbGaam4AaiaacYcacaGGOaGaam4AaiabgU caRiaadghacaGGPaGaaiilaiaadghacaGGDbGaeqiTdq2aaSbaaSqa aiqadUgagaqbaiaadUgacqGHRaWkcaWGXbaabeaakiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeikaiaabwdacaqG3aGaaeykaaaa@7BB4@ F ˜ (k,q)=2[ g 1 (q)+2 g 2 (q)] 0 L dx e iqx C k (x) C s (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia acbaGaa8hkaiaadUgacaWFSaGaamyCaiaa=LcacqGH9aqpcaWFYaGa a83waiaadEgadaWgaaWcbaGaa8xmaaqabaGccaWFOaGaamyCaiaa=L cacqGHRaWkcaWFYaGaam4zamaaBaaaleaacaWFYaaabeaakiaa=Hca caWGXbGaa8xkaiaa=1fadaWdXaqaaiaadsgacaWG4bGaamyzamaaCa aaleqabaGaamyAaiaadghacaWG4baaaOGaam4qamaaDaaaleaaceWG RbGbauaaaeaacqGHxiIkaaaabaGaa8hmaaqaaiaadYeaa0Gaey4kIi pakiaa=HcacaWG4bGaa8xkaiaadoeadaWgaaWcbaGaam4CaaqabaGc caWFOaGaamiEaiaa=Lcaaaa@5B09@ = 2π 2μ p [ g 1 (q)+2 g 2 (q)]{ iq r 0 [ μ p +ik r 0 ] }sech[π(kq) r 0 /2 μ p ]    (58) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIYaGaeqiWdahabaWaaOaaaeaacaaIYaGaeqiVd0galeqa aOWaaSbaaSqaaiaadchaaeqaaaaakiaacUfacaWGNbWaaSbaaSqaai aaigdaaeqaaOGaaiikaiaadghacaGGPaGaey4kaSIaaGOmaiaadEga daWgaaWcbaGaaGOmaaqabaGccaGGOaGaamyCaiaacMcacaGGDbWaai WaaeaadaWcaaqaaiaadMgacaWGXbGaamOCamaaBaaaleaacaaIWaaa beaaaOqaaiaacUfacqaH8oqBdaWgaaWcbaGaamiCaaqabaGccqGHRa WkcaWGPbGaam4AaiaadkhadaWgaaWcbaGaaGimaaqabaGccaGGDbaa aaGaay5Eaiaaw2haaiGacohacaGGLbGaai4yaiaadIgacaGGBbGaeq iWdaNaaiikaiaadUgacqGHsislcaWGXbGaaiykaiaadkhadaWgaaWc baGaaGimaaqabaGccaGGVaGaaGOmaiabeY7aTnaaBaaaleaacaWGWb aabeaakiaac2facaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabwda caqG4aGaaeykaaaa@6F3D@ where α q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadghaaeqaaaaa@38B6@ is determined by V(x) and the condition ( ω q vq) α q =( ω q +qv) α q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeM 8a3naaBaaaleaacaWGXbaabeaakiabgkHiTiaabAhacaqGXbGaaeyk aiabeg7aHnaaBaaaleaacaWGXbaabeaakiabg2da9iaacIcacqaHjp WDdaWgaaWcbaGaamyCaaqabaGccqGHRaWkcaqGXbGaaeODaiaabMca cqaHXoqydaqhaaWcbaGaamyCaaqaaiabgEHiQaaaaaa@4BC2@ which is required to get the factor, (1 A s + A s ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaig dacqGHsislcaWGbbWaa0baaSqaaiaadohaaeaacqGHRaWkaaGccaWG bbWaaSbaaSqaaiaadohaaeqaaOGaaiykaaaa@3DC1@ in the H ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aaaa@36D1@ in Eq.(55). Thus we find α q = iπ( χ 1 + χ 2 ) w μ p (1 s 2 ) [ M 2 ω q ] 1/2 ( ω q +qv)csch(πq r 0 /2 μ p )       (59) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadghaaeqaaOGaeyypa0ZaaSaaaeaacaWGPbGaeqiWdahc baGaa8hkaiabeE8aJnaaBaaaleaacaWFXaaabeaakiabgUcaRiabeE 8aJnaaBaaaleaacaWFYaaabeaakiaa=LcaaeaacaWG3bGaeqiVd02a aSbaaSqaaiaadchaaeqaaOGaa8hkaiaa=fdacqGHsislcaWGZbWaaW baaSqabeaacaWFYaaaaOGaa8xkaaaadaWadaqaamaalaaabaGaamyt aaqaaiaa=jdacqWIpecAcqaHjpWDdaWgaaWcbaGaamyCaaqabaaaaa GccaGLBbGaayzxaaWaaWbaaSqabeaacaWFXaGaa83laiaa=jdaaaGc caWFOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaOGaey4kaSIaamyCai aadAhacaWFPaGaci4yaiaacohacaGGJbGaamiAaiaa=HcacqaHapaC caWGXbGaamOCamaaBaaaleaacaWFWaaabeaakiaa=9cacaWFYaGaeq iVd02aaSbaaSqaaiaadchaaeqaaOGaa8xkaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabMdacaqGPa aaaa@72DF@ and W'= 2 3 μ p 2 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiaacE cacqGH9aqpdaWcaaqaaiaaikdaaeaacaaIZaaaaiabeY7aTnaaDaaa leaacaWGWbaabaGaaGOmaaaakiaadQeaaaa@3E78@ For this α q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadghaaeqaaaaa@38B6@ the | 0 ~ > ph MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaaxa cabaacbaGaa8hmaaWcbeGcbaGaa8NFaaaacqGH+aGpdaWgaaWcbaGa aeiCaiaabIgaaeqaaaaa@3C13@ in Eq.(49) is just the coherent phonon state introduced by Davydov. However, the bound state Cs(x) in Eq.(56a), unlike the unbounded state Ck(x) in Eq.(56b), is selfconsistent with the deformation. Such a self-consistent state of the intramolecular excitation and deformation forms a soliton which in the intrinsic reference frame is stationary. For the new soliton described by the state vector |ψ>= 1 2! ( A s + ) 2 |0 > ex | 0 ˜ > ph MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFai abeI8a5jabg6da+iabg2da9maalaaabaGaa8xmaaqaamaakaaabaGa a8Nmaiaa=fcaaSqabaaaaOGaa8hkaiaadgeadaqhaaWcbaGaam4Caa qaaiabgUcaRaaakiaa=LcadaahaaWcbeqaaiaa=jdaaaGccaWF8bGa a8hmaiabg6da+maaBaaaleaacaWGLbGaamiEaaqabaGccaWF8bGab8 hmayaaiaGaeyOpa4ZaaSbaaSqaaiaadchacaWGObaabeaaaaa@4BCD@ he average energy of H ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aaaa@36D1@ in Eq.(55) is <ψ| H ˜ |ψ>=2( ε 0 2J 2 v 2 4J r 0 2 ) 4 3 J μ p 2         (60) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaeq iYdKhcbaGaa8hFaiqadIeagaacaiaa=XhacqaHipqEcqGH+aGpcqGH 9aqpcaWFYaGaa8hkaiabew7aLnaaBaaaleaacaWFWaaabeaakiabgk HiTiaa=jdacaWGkbGaeyOeI0YaaSaaaeaacqWIpecAdaahaaWcbeqa aiaa=jdaaaGccaWG2bWaaWbaaSqabeaacaWFYaaaaaGcbaGaa8hnai aadQeacaWGYbWaa0baaSqaaiaa=bdaaeaacaWFYaaaaaaakiaa=Lca cqGHsisldaWcaaqaaiaa=rdaaeaacaWFZaaaaiaadQeacqaH8oqBda qhaaWcbaGaamiCaaqaaiaaikdaaaGccaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeimaiaabM caaaa@5E3E@ Evidently, the average energy of H ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aaaa@36D1@ in the soliton state |ψ> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiabeI 8a5jabg6da+aaa@39CB@ Eq.(60), is just equal to the above soliton energy Esol, or the sum of the energy of the bound state in Eq.(56a), Es, and the deformation energy of the lattice,W’, i.e., <ψ| H ~ |ψ>= E sol = E s +W MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaeq iYdKhcbaGaa8hFamaaxacabaGaamisaaWcbeGcbaGaa8NFaaaacaWF 8bGaeqiYdKNaeyOpa4Jaeyypa0deaaaaaaaaa8qacaWGfbWdamaaBa aaleaapeGaam4Caiaad+gacaWGSbaapaqabaGcpeGaeyypa0Jaamyr a8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGHRaWkcaWGxbGaai ygGaaa@4AB7@ This is an interesting result, which shows clearly that the quasi-coherent soliton formed by this mechanism is just a self-trapping state of the two excitons plus the corresponding deformation of the amino acid lattice. However, it should be noted that |ψ> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFai abeI8a5jabg6da+aaa@39D1@ is not an exact eigenstate of H ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aaaa@36D1@ owing to the presence of the terms in H ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaia aaaa@36D1@ with A k + A s MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaDa aaleaacaWGRbaabaGaey4kaScaaOaeaaaaaaaaa8qacaWGbbWdamaa BaaaleaapeGaam4CaaWdaeqaaaaa@3AFC@ and A s + A k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaDa aaleaacaWGZbaabaGaey4kaScaaOaeaaaaaaaaa8qacaWGbbWdamaa BaaaleaapeGaeyOeI0Iaam4AaaWdaeqaaaaa@3BE9@
Transition probability and decay rate of the new soliton
We now calculate the transition probability and decay rate of the quasi-coherent soliton arising from the perturbed potential by using the first-order quantum perturbation theory developed by Cottingham, et al. [14], in which the influences of the thermal and quantum effects on the properties of the soliton can be taken into account simultaneously.

For the discussion of the decay rate and lifetime of the new soliton state it is very convenlent to divide H ~ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca WGibaaleqakeaacaGG+baaaaaa@3817@ in Eq.(55) into H0+V1+V2 , where H 0 =W'+ E s A s + A s + k E k A k + A K + q ( ω q vq) b q + b q + MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaakiabg2da9iaadEfacaGGNaGaey4kaSIaamyr amaaBaaaleaacaWGZbaabeaakiaadgeadaqhaaWcbaGaam4Caaqaai abgUcaRaaakiaadgeadaWgaaWcbaGaam4CaaqabaGccqGHRaWkdaae qbqaaiaadweadaWgaaWcbaGaam4AaaqabaGccaWGbbWaa0baaSqaai aadUgaaeaacqGHRaWkaaGccaWGbbWaaSbaaSqaaiaadUeaaeqaaaqa aiaadUgaaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabl+qiOjaacI cacqaHjpWDdaWgaaWcbaGaamyCaaqabaGccqGHsislcaWG2bGaamyC aiaacMcacaWGIbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccaWGIb WaaSbaaSqaaiaadghaaeqaaOGaey4kaScaleaacaWGXbaabeqdcqGH ris5aaaa@5E98@ 1 N q ( ω q vq)( α q b q + + α q * b q )(1 A s + A s )       (61) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaWaaOaaaeaacaWGobaaleqaaaaakmaaqafabaGaeS4dHGMa aiikaiabeM8a3naaBaaaleaacaWGXbaabeaakiabgkHiTiaadAhaca WGXbGaaiykaiaacIcacqaHXoqydaWgaaWcbaGaamyCaaqabaGccaWG IbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccqGHRaWkcqaHXoqyda qhaaWcbaGaamyCaaqaaiaacQcaaaGccaWGIbWaaSbaaSqaaiaadgha aeqaaOGaaiykaiaacIcacaaIXaGaeyOeI0IaamyqamaaDaaaleaaca WGZbaabaGaey4kaScaaOGaamyqamaaBaaaleaacaWGZbaabeaakiaa cMcaaSqaaiaadghaaeqaniabggHiLdGccaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGOaGaaeOnaiaabgdacaqGPaaaaa@5F1F@ V 1 = 1 N k k q F(k,k+q,q)( b q + + b q ) A k + A k      (62) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaa baGaamOtaaWcbeaaaaGcdaaeqbqaaiaadAeacaGGOaGaam4AaiaacY cacaWGRbGaey4kaSIaamyCaiaacYcacaWGXbGaaiykaiaacIcacaWG IbWaa0baaSqaaiabgkHiTiaadghaaeaacqGHRaWkaaGccqGHRaWkca WGIbWaaSbaaSqaaiaadghaaeqaaOGaaiykaaWcbaGaam4AaiqadUga gaqbaiaadghaaeqaniabggHiLdGccaWGbbWaa0baaSqaaiqadUgaga qbaaqaaiabgUcaRaaakiaadgeadaWgaaWcbaGaam4AaaqabaGccaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeOmaiaabM caaaa@5AB7@ V 2 = 1 N kq F ~ (k,q)( b q + + b q ) ( A s + A k A s + A k ), V= V 1 + V 2      (63) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaaieaacaWFYaaabeaakiabg2da9maalaaabaGaa8xmaaqaaiaa d6eaaaWaaabuaeaadaWfGaqaaiaadAeaaSqabOqaaiaa=5haaaGaa8 hkaiaadUgacaWFSaGaa8xCaiaa=LcacaWFOaGaamOyamaaDaaaleaa cqGHsislcaWGXbaabaGaey4kaScaaOGaey4kaSIaamOyamaaBaaale aacaWGXbaabeaakiaa=LcaaSqaaiaadUgacaWGXbaabeqdcqGHris5 aOGaa8hkaiaadgeadaqhaaWcbaGaam4CaaqaaiabgUcaRaaakiaadg eadaWgaaWcbaGaam4AaaqabaGccqGHsislcaWGbbWaa0baaSqaaiaa dohaaeaacqGHRaWkaaGccaWGbbWaaSbaaSqaaiabgkHiTiaadUgaae qaaOGaa8xkaiaa=XcacaWFGaGaa8hiaiaa=zfacqGH9aqpcaWFwbWa aSbaaSqaaiaa=fdaaeqaaOGaey4kaSIaa8NvamaaBaaaleaacaWFYa aabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabAda caqGZaGaaeykaaaa@676F@ where H0 describes the relevant quasi-particle excitations in the protein. This is a soliton together with phonons relative to the distorted amino acid lattice. The resulting delocalized excitations belongs to an exciton-like band with phonons relative to a uniform lattice. The bottom of the band of the latter is at the energy 4J μ p 2 /3 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadQ eacqaH8oqBdaqhaaWcbaGaamiCaaqaaiaaikdaaaGccaGGVaGaaG4m aaaa@3C90@ relative to the soliton, in which the topological stability associated with removing the lattice distortion is included.

We now calculate the decay rate of the new soliton along the following lines by using Eq.(61) and V2 in Eq.(63) and quantum perturbation theory. Firstly, we compute a more general formula for the decay rate of the soliton containing n quanta in the system in which the three terms contained in Eq.(40a) is replaced by (n+1) terms of the expression of a coherent state exp{ n [ φ n ( t ) B n + φ n ( t ) B n ] }| 0 > ex MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacI hacaGGWbWaaiWaaeaadaaeqbqaamaadmaabaGaeqOXdO2aaSbaaSqa aiaad6gaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaamOqam aaDaaaleaacaWGUbaabaGaey4kaScaaOGaeyOeI0IaeqOXdO2aa0ba aSqaaiaad6gaaeaacqGHxiIkaaGcdaqadaqaaiaadshaaiaawIcaca GLPaaacaWGcbWaaSbaaSqaaiaad6gaaeqaaaGccaGLBbGaayzxaaaa leaacaWGUbaabeqdcqGHris5aaGccaGL7bGaayzFaaWaaqqaaeaaca aIWaGaeyOpa4ZaaSbaaSqaaiaadwgacaWG4baabeaaaOGaay5bSdaa aa@56F6@ Finally we find out the decay rate of the new soliton with two-quanta. In such a case H0 is chosen such the ground state, |n> has energy W'+n E s ' MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiaacE cacqGHRaWkcaWGUbGaamyramaaDaaaleaacaWGZbaabaGaai4jaaaa aaa@3BEB@ in the subspace of excitation number equal to n, i.e., <n| i B i + B i |n>=<nl( A s + A s + k A k + A k )ln>=n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam OBaiaacYhadaaeqbqaaiaadkeadaqhaaWcbaGaamyAaaqaaiabgUca RaaakiaadkeadaWgaaWcbaGaamyAaaqabaacbaGccaWFGaGaa8hFai aa=5gacqGH+aGpcqGH9aqpcqGH8aapcaWFUbGaa8hBaiaa=HcacaWF bbWaa0baaSqaaiaa=nhaaeaacqGHRaWkaaGccaWFbbWaaSbaaSqaai aa=nhaaeqaaOGaey4kaSYaaabuaeaacaWFbbWaa0baaSqaaiaa=Tga aeaacqGHRaWkaaGccaWGbbWaaSbaaSqaaiaadUgaaeqaaaqaaiaa=T gaaeqaniabggHiLdGccaWFPaGaa8hBaiaa=5gacqGH+aGpcqGH9aqp caWFUbaaleaacaWGPbaabeqdcqGHris5aaaa@5BBA@ In this subspace the eigenstates have the simple form |n-m,k1k2… km, {nq}> 1 (nm)! ( A S + ) nm A k 1 + A k 2 + A k m + |0 > ex Π q ( d q + ) n q n q ! 1 0 ˜ > ph nm        (64) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaie aacaWFXaaabaWaaOaaaeaacaWFOaGaamOBaiabgkHiTiaad2gacaWF PaGaa8xiaaWcbeaaaaGccaWFOaGaamyqamaaDaaaleaacaWGtbaaba Gaey4kaScaaOGaa8xkamaaCaaaleqabaGaamOBaiabgkHiTiaad2ga aaGccaWGbbWaa0baaSqaaiaadUgadaWgaaadbaGaa8xmaaqabaaale aacqGHRaWkaaGccaWGbbWaa0baaSqaaiaadUgadaWgaaadbaGaa8Nm aaqabaaaleaacqGHRaWkaaGccqWIVlctcaWGbbWaa0baaSqaaiaadU gadaWgaaadbaGaamyBaaqabaaaleaacqGHRaWkaaGccaWF8bGaa8hm aiabg6da+maaBaaaleaacaWGLbGaamiEaaqabaGcdaWfqaqaaiabfc 6aqbWcbaGaamyCaaqabaGcdaWcaaqaaiaa=HcacaWGKbWaa0baaSqa aiaadghaaeaacqGHRaWkaaGccaWFPaWaaWbaaSqabeaacaWGUbWaaS baaWqaaiaadghaaeqaaaaaaOqaamaakaaabaGaamOBamaaBaaaleaa caWGXbaabeaakiaa=fcaaSqabaaaaOGaaGymaiqaicdagaacaiabg6 da+maaDaaaleaacaWGWbGaamiAaaqaaiaad6gacqGHsislcaWGTbaa aOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabI cacaqG2aGaaeinaiaabMcaaaa@6FE4@ where d q = b q + m n 1 N α q = a q nm n 1 N α q (m≤n, n and m are all intgers)       (65) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGXbaabeaakiabg2da9iaadkgadaWgaaWcbaGaamyCaaqa baGccqGHRaWkdaWcaaqaaiaad2gaaeaacaWGUbaaamaalaaabaacba Gaa8xmaaqaamaakaaabaGaamOtaaWcbeaaaaGccqaHXoqydaWgaaWc baGaamyCaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaadghaaeqaaO GaeyOeI0YaaSaaaeaacaWGUbGaeyOeI0IaamyBaaqaaiaad6gaaaWa aSaaaeaacaWFXaaabaWaaOaaaeaacaWGobaaleqaaaaakiabeg7aHn aaBaaaleaacaWGXbaabeaakiaabIcaqaaaaaaaaaWdbiaab2gacaqG JcGaaeOBaiaabYcacaqGGaGaaeOBaiaabccacaqGHbGaaeOBaiaabs gacaqGGaGaaeyBaiaabccacaqGHbGaaeOCaiaabwgacaqGGaGaaeyy aiaabYgacaqGSbGaaeiiaiaabMgacaqGUbGaaeiDaiaabEgacaqGLb GaaeOCaiaabohapaGaaeykaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGOaGaaeOnaiaabwdacaqGPaaaaa@6FE3@ with d q | 0 ˜ > ph nm =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGKbWdamaaBaaaleaapeGaamyCaaWdaeqaaOGaaiiFaiqaicda gaacaiabg6da+maaDaaaleaacaWGWbGaamiAaaqaaiaad6gacqGHsi slcaWGTbaaaOGaeyypa0JaaGimaaaa@41D4@ The corresponding energy of the systems is E nm; k 1 ... k m 1 ;{ n q } (0) =(1 (m/n) 2 )W'+(nm) E s + j=1 m E k 1 + q ( ω q vq) n q      (66) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaDa aaleaacaWGUbGaeyOeI0IaamyBaGqaaiaa=TdacaWGRbWaaSbaaWqa aiaa=fdaaeqaaSGaa8Nlaiaa=5cacaWFUaGaam4AamaaBaaameaaca WGTbWaaSbaaeaacaWFXaaabeaaaeqaaSGaa83oaiaa=ThacaWGUbWa aSbaaWqaaiaadghaaeqaaSGaa8xFaaqaaiaa=HcacaWFWaGaa8xkaa aakiabg2da9iaa=HcacaWFXaGaeyOeI0Iaa8hkaiaad2gacaWFVaGa amOBaiaa=LcadaahaaWcbeqaaiaa=jdaaaGccaWFPaGaam4vaiaacE cacqGHRaWkcaWFOaGaamOBaiabgkHiTiaad2gacaWFPaGabmyrayaa faWaaSbaaSqaaiaadohaaeqaaOGaey4kaSYaaabCaeaaceWGfbGbau aadaqhaaWcbaGaam4AamaaBaaameaacaWFXaaabeaaaSqaaaaakiab gUcaRmaaqafabaGaeS4dHGMaa8hkaiabeM8a3naaBaaaleaacaWGXb aabeaakiabgkHiTiaadAhacaWGXbGaa8xkaiaad6gadaWgaaWcbaGa amyCaaqabaaabaGaamyCaaqab0GaeyyeIuoaaSqaaiaadQgacqGH9a qpcaWFXaaabaGaamyBaaqdcqGHris5aOGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGOaGaaeOnaiaabAdacaqGPaaaaa@7796@ E s MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaadohaaeqaaaaa@37EF@ is the energy of a bound state with one exciton, E k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaadUgaaeqaaaaa@37E7@ is the energy of the unbound(delocalized) state with one exciton. When m=0 the excitation state is a n-type soliton plus phonons relative to the chain with the deformation corresponding to the n-type soliton. For m=n the excited states are delocalized and the phonons are relative to a chain without any deformation. Furthermore except for small k, the delocalized states approximate ordinary excitons. Thus the decay of the soliton is just a transition from the initial state with the n-type soliton plus the new phonons: |n>= 1 n! Π q ( b q + ) n q ( n q !) 1/2 ( A s + ) n |0 > ex | 0 ~ > ph        (67) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFai aa=5gacqGH+aGpcqGH9aqpdaWcaaqaaiaa=fdaaeaadaGcaaqaaiaa =5gacaWFHaaaleqaaaaakmaaxababaGaeuiOdafaleaacaWGXbaabe aakmaalaaabaGaa8hkaiaadkgadaqhaaWcbaGaamyCaaqaaiabgUca Raaakiaa=LcadaahaaWcbeqaaiaad6gadaWgaaqaaiaadghaaeqaaa aaaOqaaiaa=HcacaWGUbWaaSbaaSqaaiaadghaaeqaaOGaa8xiaiaa =LcadaahaaWcbeqaaiaa=fdacaWFVaGaa8NmaaaaaaGccaWFOaGaam yqamaaDaaaleaacaWGZbaabaGaey4kaScaaOGaa8xkamaaCaaaleqa baGaa8NBaaaakiaa=XhacaWFWaGaeyOpa4ZaaSbaaSqaaiaadwgaca WG4baabeaakiaa=XhadaWfGaqaaiaa=bdaaSqabOqaaiaa=5haaaGa eyOpa4ZaaSbaaSqaaiaadchacaWGObaabeaakiaa=bcacaWFGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG 2aGaae4naiaabMcaaaa@64E5@ With corresponding energy Es{nq} =w+n E s + q ( ω q vq) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaadohaaeqaaOGaey4kaSYaaabuaeaacqWIpecAieaa caWFOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaOGaeyOeI0IaamODai aadghacaWFPaaaleaacaWGXbaabeqdcqGHris5aaaa@4450@ nq to the final state with delocalized excitons and the original phonons: |αk>= Π q ( a q + ) n q n q ! |0 > ph ( A k + ) n |0 > ex       (68) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8hFai abeg7aHjaadUgacqGH+aGpcqGH9aqpdaWfqaqaaiabfc6aqbWcbaGa amyCaaqabaGcdaWcaaqaaiaa=HcacaWGHbWaa0baaSqaaiaadghaae aacqGHRaWkaaGccaWFPaWaaWbaaSqabeaacaWGUbWaaSbaaeaacaWG XbaameqaaaaaaOqaamaakaaabaGaamOBamaaDaaaleaacaWGXbaaba aaaOGaa8xiaaWcbeaaaaGccaWF8bGaa8hmaiabg6da+maaBaaaleaa caWGWbGaamiAaaqabaGccaWFOaGaamyqamaaDaaaleaacaWGRbaaba Gaey4kaScaaOGaa8xkamaaCaaaleqabaGaa8NBaaaakiaacYhacaaI WaGaeyOpa4ZaaSbaaSqaaiaadwgacaWG4baabeaakiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeioaiaabMca aaa@5D6D@ with corresponding energy Ek{nq}=n E k + q ( ω q vq) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa Waa0baaSqaaiaadUgaaeaaaaGccqGHRaWkdaaeqbqaaiabl+qiOjaa cIcacqaHjpWDdaWgaaWcbaGaamyCaaqabaGccqGHsislcaWG2bGaam yCaiaacMcaaSqaaiaadghaaeqaniabggHiLdaaaa@4446@ nq caused by the part, V2, in the perturbation interaction V. In this case, the initial phonon distribution will be taken to be at thermal equilibrium. The probability of the above transitions in lowest order perturbation theory is given by W ¯ = 1 2 0 t d t 0 t d t { α k l P l (ph) <n|exp( i H 0 t ) V 2 exp( i H 0 t ) |α k >        (69) <α k |exp( i H 0 t ) V 2 exp( i H 0 t h ) |n> } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaceWGxb GbaebacqGH9aqpdaWcaaqaaGqaaiaa=fdaaeaacqWIpecAdaahaaWc beqaaiaa=jdaaaaaaOWaa8qmaeaacaWGKbGabmiDayaafaWaa8qmae aacaWGKbGabmiDayaagaaaleaacaWFWaaabaGaamiDaaqdcqGHRiI8 aOWaaiqaaeaadaaeqbqaamaaqafabaGaamiuamaaDaaaleaacaWGSb aabaGaa8hkaiaadchacaWGObGaa8xkaaaakiabgYda8iaa=5gacaWF 8bGaciyzaiaacIhacaGGWbWaaeWaaeaadaWcaaqaaiaadMgacaWGib WaaSbaaSqaaiaa=bdaaeqaaOGabmiDayaagaaabaGaeS4dHGgaaaGa ayjkaiaawMcaaiaadAfadaWgaaWcbaGaa8NmaaqabaGcciGGLbGaai iEaiaacchadaqadaqaamaalaaabaGaeyOeI0IaamyAaiaadIeadaWg aaWcbaGaa8hmaaqabaGcceWG0bGbayaaaeaacqWIpecAaaaacaGLOa GaayzkaaaaleaacaWGSbaabeqdcqGHris5aaWcbaGaeqySdeMabm4A ayaafaaabeqdcqGHris5aaGccaGL7baacaWF8bGaeqySdeMabm4Aay aafaGaeyOpa4daleaacaWFWaaabaGaamiDaaqdcqGHRiI8aOGaeyyX ICTaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabI cacaqG2aGaaeyoaiaabMcaaeaacaWFGaGaa8hiaiaa=bcacaWFGaGa a8hiaiabgYda8iabeg7aHjqa=Tgagaqbaiaa=XhacaWFLbGaa8hEai aa=bhadaqadaqaamaalaaabaGaa8xAaiaa=HeadaWgaaWcbaGaa8hm aaqabaGcceWG0bGbauaaaeaacqWIpecAaaaacaGLOaGaayzkaaGaam OvamaaBaaaleaacaWFYaaabeaakiGacwgacaGG4bGaaiiCamaabmaa baWaaSaaaeaacqGHsislcaWGPbGaamisamaaBaaaleaacaWFWaaabe aakiqadshagaqbaaqaaiaadIgaaaaacaGLOaGaayzkaaWaaiGaaeaa caWF8bGaamOBaiabg6da+aGaayzFaaaaaaa@9BA5@ We should calculate the transition probability of the soliton resulting from the perturbed potential, (V1+V2), at first-order in perturbation theory. Following Cottingham and Schweitzer [14], we estimate only the transition from the soliton state to delocalized exciton states caused by the potential V2, which can satisfactorily be treated by means of perturbation theory since the coefficient F ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia aaaa@36CF@ (k,q) defined by Eq.(58) is proportional to an integral over the product of the localized state and a delocalized state, and therefore is of order 1/ N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGobaaleqaaaaa@36E3@ . The V1 term in the Hamiltonian is an interaction between the delocalized excitons and the phonons. The main effect of V1 is to modify the spectrum of the delocalized excitatons in the weak coupling limit (Jμp/ KBT0 << 1 , the definition of T0 is given below). As a result the delocalized excitons and phonons will have their energies shifted and also have finite lifetimes. These effects are ignored in our calculation since they are only of second order in V1.

The sum over l in Eq.(69) indicates a sum over an initial set of occupation numbers for phonons relative to the distorted amino acid lattice with probability distribution , which is taken to be the thermal equilibrium distribution for a given temperature T. Since P l ph MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaDa aaleaacaWGSbaabaGaamiCaiaadIgaaaaaaa@39CA@ ,which is taken to be the thermal equilibrium distribution for a given temperature T. Since e i H 0 t |n, { n q }>=exp{-i(W'+n E q )t/-i q ( ω q -qv) b q + b q t}|n,{ n q }> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaeyOeI0IaamyAaiaadIeadaWgaaqaaGqaaiaa=bdaaWqa baWccaWG0baaaOGaaiiFaiaad6gacaWFSaGaa8hiaiaa=ThacaWFUb WaaSbaaSqaaiaa=fhaaeqaaOGaa8xFaiabg6da+iabg2da9iaa=vga caWF4bGaa8hCaiaa=ThacaWFTaGaa8xAaiaa=HcacaWFxbGaa83jai abgUcaRiaa=5gaceWFfbGbauaadaWgaaWcbaGaa8xCaaqabaGccaWF PaGaa8hDaiaa=9cacqWIpecAcaWFTaGaa8xAamaaqafabaGaa8hkai abeM8a3naaBaaaleaacaWFXbaabeaakiaa=1cacaWFXbGaa8NDaiaa =LcaaSqaaiaa=fhaaeqaniabggHiLdGccaWFIbWaa0baaSqaaiaa=f haaeaacqGHRaWkaaGccaWFIbWaaSbaaSqaaiaa=fhaaeqaaOGaa8hD aiaa=1hacaWF8bGaa8NBaiaa=XcacaWF7bGaa8NBamaaBaaaleaaca WFXbaabeaakiaa=1hacqGH+aGpaaa@6E0D@ and e i H 0 t |n1, { n q }>=exp{ -i[(1 1 n 2 )W'+(n1) E s + E k ] t/-i q ( ω q qv) d q + d q t }|n1,{ n q }> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAaiaadIeadaWgaaadbaacbaGaa8hmaaqabaWccaWG 0baaaOGaa8hFaiaad6gacqGHsislcaWFXaGaa8hlaiaa=bcacaWF7b Gab8NBayaafaWaa0baaSqaaiaa=fhaaeaaaaGccaWF9bGaeyOpa4Ja eyypa0Jaa8xzaiaa=HhacaWFWbWaaiqaaeaacaWFTaGaa8xAaiaa=T facaWFOaGaa8xmaiabgkHiTmaalaaabaGaa8xmaaqaaiaa=5gadaah aaWcbeqaaiaa=jdaaaaaaOGaa8xkaiaa=DfacaWFNaGaey4kaSIaa8 hkaiaa=5gacqGHsislcaWFXaGaa8xkaiqa=veagaqbamaaBaaaleaa caWFZbaabeaakiabgUcaRiqa=veagaqbamaaBaaaleaacaWFRbaabe aakiaa=1faaiaawUhaaiaadshacaWFVaGaeS4dHGMaa8xlaiaa=Lga daaeqbqaaiaa=HcacqaHjpWDdaWgaaWcbaGaamyCaaqabaGccqGHsi slcaWGXbGaamODaiaa=LcacaWFKbWaa0baaSqaaiaadghaaeaacqGH RaWkaaaabaGaa8xCaaqab0GaeyyeIuoakmaaciaabaGaamizamaaDa aaleaacaWGXbaabaaaaOGaamiDaaGaayzFaaGaa8hFaiaad6gacqGH sislcaWFXaGaa8hlaiaa=ThaceWFUbGbauaadaWgaaWcbaGaa8xCaa qabaGccaWF9bGaeyOpa4daaa@7CC2@ where d q =b q + 1 n 1 N α q , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG3bGaaeiAaiaabwgacaqGYbGaaeyzaiaabckacaqGKbWdamaa BaaaleaapeGaaeyCaaWdaeqaaOWdbiaab2dacaqGIbWdamaaBaaale aapeGaaeyCaaWdaeqaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaamOB aaaadaWcaaqaaiaaigdaaeaadaGcaaqaaiaad6eaaSqabaaaaOGaeq ySde2aaSbaaSqaaiaadghaaeqaaOGaaiilaaaa@490C@ using the explicit form for V2 and the fact that the sum over states| k α, { n q }> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaafa GaeqySdeMaaiilaiaabccaieaacaWF7bGabmOBayaafaWaaSbaaSqa aiaadghaaeqaaOGaa8xFaiabg6da+aaa@3F19@ contains a complete set of phonons for each values of k′, one can rewrite W ¯ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara aaaa@36E9@ as   W ¯ = 1 2 π 2 2n μ 1 N 2 k k k [ g 1 (k)+2 g 2 (k)] [ g 1 ( k )+2 g 2 ( k )] (k r 0 )( k r 0 ) (n μ 1 ) 2 + ( k r 0 ) 2 Sech[ π r 0 2n μ 1 (k k ) ] sech[ π r 0 2n μ 1 ( k k ) ] 0 t d t 0 t d t { exp[ i ( n( n 2 2 3 n) μ 1 2 J+nJ ( k r 0 ) 2 )( t t ) ]                               (70)  <<exp[i q ( ω q qv) b q + b q ( t t ) ]( b k + + b k ) exp[i q ( ω q qv) a q + a q ( t t )( b k + + b k )>> } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGGa Gabm4vayaaraGaeyypa0ZaaSaaaeaaieaacaWFXaaabaGaeS4dHG2a aWbaaSqabeaacaWFYaaaaaaakmaalaaabaGaeqiWda3aaWbaaSqabe aacaWFYaaaaaGcbaGaa8Nmaiaa=5gacqaH8oqBdaWgaaWcbaGaaGym aaqabaGccaWGobWaaWbaaSqabeaacaWFYaaaaaaakmaaqafabaWaaa buaeaadaaeqbqaaiaa=TfacaWGNbWaa0baaSqaaiaa=fdaaeaacqGH xiIkaaGccaWFOaGaam4Aaiaa=LcacqGHRaWkcaWFYaGaam4zamaaDa aaleaacaWFYaaabaGaey4fIOcaaOGaa8hkaiaadUgacaWFPaGaa8xx aaWcbaGabm4AayaagaaabeqdcqGHris5aaWcbaGabm4Aayaafaaabe qdcqGHris5aOGaa8hiaaWcbaGaam4Aaaqab0GaeyyeIuoakiaa=Tfa caWGNbWaaSbaaSqaaiaa=fdaaeqaaOGaa8hkaiqadUgagaGbaiaa=L cacqGHRaWkcaWFYaGaam4zamaaBaaaleaacaWFYaaabeaakiaa=Hca ceWGRbGbayaacaWFPaGaa8xxaiaa=bcadaWcaaqaaiaa=HcacaWFRb Gaa8NCamaaBaaaleaacaWFWaaabeaakiaa=LcacaWFOaGab83Aayaa gaGaa8NCamaaBaaaleaacaWFWaaabeaakiaa=LcaaeaacaGGOaGaam OBaiabeY7aTnaaDaaaleaacaaIXaaabaaaaOGaaiykamaaCaaaleqa baGaaGOmaaaakiabgUcaRiaa=HcaceWGRbGbauaacaWGYbWaaSbaaS qaaiaa=bdaaeqaaOGaa8xkamaaCaaaleqabaGaa8NmaaaaaaGccaWG tbGaamyzaiaadogacaWGObWaamWaaeaadaWcaaqaaiabec8aWjaadk hadaWgaaWcbaGaa8hmaaqabaaakeaacaWFYaGaa8NBaiabeY7aTnaa BaaaleaacaaIXaaabeaaaaGccaWFOaGaam4AaiabgkHiTiqadUgaga qbaiaa=LcaaiaawUfacaGLDbaacqGHflY1aeaacaWFGaGaa8hiaiGa cohacaGGLbGaai4yaiaadIgadaWadaqaamaalaaabaGaeqiWdaNaam OCamaaBaaaleaacaWFWaaabeaaaOqaaiaa=jdacaWFUbGaeqiVd02a aSbaaSqaaiaaigdaaeqaaaaakiaa=HcaceWGRbGbayaacqGHsislce WGRbGbauaacaWFPaaacaGLBbGaayzxaaGaa8hiamaapedabaGaamiz aiqadshagaqbaaWcbaGaa8hmaaqaaiaadshaa0Gaey4kIipakmaape dabaGaamizaiqadshagaGbamaaceaabaGaciyzaiaacIhacaGGWbWa amWaaeaadaWcaaqaaiabgkHiTiaadMgaaeaacqWIpecAaaWaaeWaae aacaWGUbGaaiikaiaad6gadaahaaWcbeqaaiaaikdaaaGccqGHsisl daWcaaqaaiaaikdaaeaacaaIZaaaaiaad6gacaGGPaGaeqiVd02aa0 baaSqaaiaaigdaaeaacaaIYaaaaOGaamOsaiabgUcaRiaad6gacaWG kbGaa8hkaiqadUgagaqbaiaadkhadaWgaaWcbaGaa8hmaaqabaGcca WFPaWaaWbaaSqabeaacaWFYaaaaaGccaGLOaGaayzkaaGaa8hkaiqa dshagaqbaiabgkHiTiqadshagaGbaiaa=LcaaiaawUfacaGLDbaacq GHflY1aiaawUhaaaWcbaGaa8hmaaqaaiaadshaa0Gaey4kIipakiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaeim aiaabMcaaeaacaqGGaGaeyipaWJaeyipaWJaciyzaiaacIhacaGGWb Gaa83waiaadMgadaaeqbqaaiaa=HcacqaHjpWDdaWgaaWcbaGaamyC aaqabaGccqGHsislcaWGXbGaamODaiaa=LcadaWacaqaaiaadkgada qhaaWcbaGaamyCaaqaaiabgUcaRaaakiaadkgadaWgaaWcbaGaamyC aaqabaGccaWFOaGabmiDayaafaGaeyOeI0IabmiDayaagaGaa8xkaa GaayzxaaGaa8hkaiaadkgadaqhaaWcbaGaam4AaaqaaiabgUcaRaaa kiabgUcaRiaadkgadaWgaaWcbaGaeyOeI0Iaam4AaaqabaaabaGaam yCaaqab0GaeyyeIuoakiaa=LcadaGacaqaaiGacwgacaGG4bGaaiiC aiaa=TfacaWGPbWaaabuaeaacaWFOaGaeqyYdC3aaSbaaSqaaiaadg haaeqaaOGaeyOeI0IaamyCaiaadAhacaWFPaGaamyyamaaDaaaleaa caWGXbaabaGaey4kaScaaOGaamyyamaaBaaaleaacaWGXbaabeaaki aa=HcaceWG0bGbauaacqGHsislceWG0bGbayaacaWFPaGaa8hkaiaa =jgadaqhaaWcbaGaeyOeI0Iabm4AayaagaaabaGaey4kaScaaOGaey 4kaSIaamOyamaaBaaaleaaceWGRbGbayaaaeqaaaqaaiaadghaaeqa niabggHiLdGccaWFPaGaeyOpa4JaeyOpa4dacaGL9baaaaaa@314E@ where g 1 (k)+2 g 2 (k)=2 χ 1 ( 2M ω k ) 1/2 [A(cos( r 0 k)1)+i(A+1)sin( r 0 k)]2i(A+1)( r 0 k) χ 1 ( 2M ω k ) 1/2 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaaabeaakiaacIcacaWGRbGaaiykaiabgUcaRiaaikda caWGNbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadUgacaGGPaGaey ypa0JaaGOmaiabeE8aJnaaBaaaleaacaaIXaaabeaakiaacIcadaWc aaqaaiabl+qiObqaaiaaikdacaWGnbGaeqyYdC3aaSbaaSqaaiaadU gaaeqaaaaakiaacMcadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaa kiaacUfacaWGbbGaaiikaiGacogacaGGVbGaai4CaiaacIcacaWGYb WaaSbaaSqaaiaaicdaaeqaaOGaam4AaiaacMcacqGHsislcaaIXaGa aiykaiabgUcaRiaadMgacaGGOaGaamyqaiabgUcaRiaaigdacaGGPa Gaci4CaiaacMgacaGGUbGaaiikaiaadkhadaWgaaWcbaGaaGimaaqa baGccaWGRbGaaiykaiaac2facqGHijYUcaaIYaGaamyAaiaacIcaca WGbbGaey4kaSIaaGymaiaacMcacaGGOaGaamOCamaaBaaaleaacaaI WaaabeaakiaadUgacaGGPaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaO GaaiikamaalaaabaGaeS4dHGgabaGaaGOmaiaad2eacqaHjpWDdaWg aaWcbaGaam4AaaqabaaaaOGaaiykamaaCaaaleqabaGaaGymaiaac+ cacaaIYaaaaOGaaiilaaaa@7EEF@ μ 1 = χ 1 2 (1+ A 2 ) ω(1 s 2 )J ,A= χ 2 / χ 1           (71) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaHhpWydaqhaaWc baGaaGymaaqaaiaaikdaaaGccaGGOaGaaGymaiabgUcaRiaadgeada ahaaWcbeqaaiaaikdaaaGccaGGPaaabaGaeqyYdCNaaiikaiaaigda cqGHsislcaWGZbWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaadQeaaa GaaiilaiaadgeacqGH9aqpcqaHhpWydaWgaaWcbaGaaGOmaaqabaGc caGGVaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bIcacaqG3aGaaeymaiaabMcaaaa@5AF1@ here A is a new parameter introduced to describe the rate between the new nonlinear interaction term and the one in the Davydov’s model.

To estimate the lifetime of the soliton we are interested in the long-time behavior of d w ¯ dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGabm4DayaaraaabaGaamizaiaadshaaaaaaa@39E4@ . By straightforward calculation, the average transition probability or decay rate of the soliton is given by Γ n = lim t d W ¯ dt = 4 [ π 2 2n μ 1 N 2 ] k k k'' [ [ g 1 * (k)+2 g 2 * (k)] [ g 1 ( k )+2 g 2 ( k )] (k r 0 )( k r 0 ) (n μ 1 ) 2 + ( k r 0 ) 2 sech[ π r 0 2n μ 1 (k k ) ] sech[ π r 0 2n μ 1 ( k k ) ]Re{ 0 dtexp[ i ( n( n 2 2 3 n) μ 1 2 J+nJ ( k r 0 ) 2 )t ] <<exp[i q ( ω q qv) b q + b q t]( b k + + b k ) exp[i q ( ω q qv) a q + a q t]( b k + + b k )>> } } = 4 2 π 2 2n μ 1 N 2 k k k'' { [ g 1 * (k)+2 g 2 * (k)] [ g 1 ( k )+2 g 2 ( k )] (k r 0 )( k r 0 ) (n μ 1 ) 2 + ( k r 0 ) 2 sech[ π r 0 (k k ) 2n μ 1 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqqHto WrdaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWfqaqaaiGacYgacaGG 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=HcacaWFRbGaa8xkaiaa=1faaiaawUhaaaWcbaGaam4AaiqadUgaga qbaiaadUgacaWFNaGaa83jaaqab0GaeyyeIuoakiaa=TfacaWGNbWa aSbaaSqaaiaa=fdaaeqaaOGaa8hkaiqadUgagaGbaiaa=LcacqGHRa WkcaWFYaGaam4zamaaBaaaleaacaWFYaaabeaakiaa=HcaceWGRbGb ayaacaWFPaGaa8xxamaalaaabaGaa8hkaiaadUgacaWGYbWaaSbaaS qaaiaa=bdaaeqaaOGaa8xkaiaa=HcaceWGRbGbayaacaWGYbWaaSba aSqaaiaa=bdaaeqaaOGaa8xkaaqaaiaacIcacaWGUbGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaaiykamaaDaaaleaaaeaacaaIYaaaaOGa ey4kaSIaa8hkaiqadUgagaqbaiaadkhadaWgaaWcbaGaa8hmaaqaba GccaWFPaWaaWbaaSqabeaacaWFYaaaaaaakiGacohacaGGLbGaai4y aiaadIgadaWadaqaamaalaaabaGaeqiWdaNaamOCamaaBaaaleaaca WFWaaabeaakiaa=HcacaWGRbGaeyOeI0Iabm4AayaafaGaa8xkaaqa aiaa=jdacaWFUbGaeqiVd02aaSbaaSqaaiaaigdaaeqaaaaaaOGaay 5waiaaw2faaiabgwSixdaaaa@70D0@ sech[ π r 0 2n μ 1 ( k k ) ]Re 0 dtU(k, k t)exp [ i ( n( n 2 2 3 n) μ 1 2 J+nJ ( k r 0 ) 2 )t ] }  (72) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaaci GGZbGaaiyzaiaacogacaWGObWaamWaaeaadaWcaaqaaiabec8aWjaa dkhadaWgaaWcbaacbaGaa8hmaaqabaaakeaacaWFYaGaa8NBaiabeY 7aTnaaBaaaleaacaaIXaaabeaaaaGccaWFOaGabm4AayaagaGaeyOe I0Iabm4AayaafaGaa8xkaaGaay5waiaaw2faaiGackfacaGGLbWaa8 qmaeaacaWGKbGaamiDaiaadwfacaWFOaGaam4Aaiaa=XcaceWGRbGb ayaacaWG0bGaa8xkaiGacwgacaGG4bGaaiiCamaadeaabaGaeyOeI0 YaaSaaaeaacaWGPbaabaGaeS4dHGgaaaGaay5waaWaamGaaeaadaqa daqaaiaad6gacaGGOaGaamOBamaaCaaaleqabaGaaGOmaaaakiabgk HiTmaalaaabaGaaGOmaaqaaiaaiodaaaGaamOBaiaacMcacqaH8oqB daqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWGkbGaey4kaSIaamOBai aadQeacaWFOaGabm4AayaafaGaamOCamaaBaaaleaacaWFWaaabeaa kiaa=LcadaahaaWcbeqaaiaa=jdaaaaakiaawIcacaGLPaaacaWG0b aacaGLDbaaaSqaaiaa=bdaaeaacqGHEisPa0Gaey4kIipaaOGaayzF aaGaaeiiaiaabccacaqGOaGaae4naiaabkdacaqGPaaaaa@78A7@ where the thermal average is U(k, k ,t)=<<exp[i q ( ω q qv) b q + b q t]( b k + + b k )exp[i q ( ω q qv) a q + a q t]( b k + + b k )>> MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaiab g2da9iabgYda8iabgYda8iGacwgacaGG4bGaaiiCaiaa=TfacaWGPb WaaabuaeaacaWFOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaOGaeyOe I0IaamyCaiaadAhacaWFPaGaamOyamaaDaaaleaacaWGXbaabaGaey 4kaScaaOGaamOyamaaDaaaleaacaWGXbaabaaaaOGaa8hDaiaa=1fa caWFOaGaamOyamaaDaaaleaacaWGRbaabaGaey4kaScaaOGaey4kaS IaamOyamaaBaaaleaacqGHsislcaWGRbaabeaaaeaacaWGXbaabeqd cqGHris5aOGaa8xkaiGacwgacaGG4bGaaiiCaiaa=TfacqGHsislca WGPbWaaabuaeaacaWFOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaOGa eyOeI0IaamyCaiaadAhacaWFPaGaamyyamaaDaaaleaacaWGXbaaba Gaey4kaScaaOGaamyyamaaBaaaleaacaWGXbaabeaakiaadshacaWF DbGaa8hkaiaadkgadaqhaaWcbaGaeyOeI0Iabm4AayaagaaabaGaey 4kaScaaOGaey4kaSIaamOyamaaBaaaleaaceWGRbGbayaaaeqaaaqa aiaadghaaeqaniabggHiLdGccaWFPaGaeyOpa4JaeyOpa4daaa@7E43@ with <<A>>=Tr{Aexp[β q ( ω q qv) b q + b q ]}/Tr{exp[β q ( ω q qv) b q + b q ]} MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaey ipaWJaamyqaiabg6da+iabg6da+iabg2da9iaadsfacaWGYbGaai4E aiaadgeaciGGLbGaaiiEaiaacchacaGGBbGaeyOeI0IaeqOSdi2aaa buaeaacqWIpecAcaGGOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaOGa eyOeI0IaamyCaiaadAhacaGGPaGaamOyamaaDaaaleaacaWGXbaaba Gaey4kaScaaOGaamOyamaaBaaaleaacaWGXbaabeaaaeaacaWGXbaa beqdcqGHris5aOGaaiyxaiaac2hacaGGVaGaamivaiaadkhacaGG7b GaciyzaiaacIhacaGGWbGaai4waiabgkHiTiabek7aInaaqafabaGa eS4dHGMaaiikaiabeM8a3naaBaaaleaacaWGXbaabeaakiabgkHiTi aadghacaWG2bGaaiykaiaadkgadaqhaaWcbaGaamyCaaqaaiabgUca RaaakiaadkgadaWgaaWcbaGaamyCaaqabaaabaGaamyCaaqab0Gaey yeIuoakiaac2facaGG9baaaa@7424@ Tr{Aexp[β q ( ω q qv) b q + b q ]}/ Z ph        (73) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaadk hacaGG7bGaamyqaiGacwgacaGG4bGaaiiCaiaacUfacqGHsislcqaH YoGydaaeqbqaaiabl+qiOjaacIcacqaHjpWDdaWgaaWcbaGaamyCaa qabaGccqGHsislcaWGXbGaamODaiaacMcacaWGIbWaa0baaSqaaiaa dghaaeaacqGHRaWkaaGccaWGIbWaaSbaaSqaaiaadghaaeqaaaqaai aadghaaeqaniabggHiLdGccaGGDbGaaiyFaiaac+cacaWGAbWaaSba aSqaaiaadchacaWGObaabeaakiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGOaGaae4naiaabodacaqGPaaaaa@5D0A@ and   Z ph = q (1exp[β( ω q qv)]} 1 ,  (β= 1 K B T ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGHbGaaeOBaiaabsgacaGGGcGaaiiOaiaadQfapaWaaSbaaSqa a8qacaWGWbGaamiAaaWdaeqaaOWdbiabg2da98aadaWfqaqaaiabg+ GivdWcbaGaamyCaaqabaGccaGGOaGaaGymaiabgkHiTiGacwgacaGG 4bGaaiiCaiaacUfacqGHsislcqaHYoGycqWIpecAcaGGOaGaeqyYdC 3aaSbaaSqaaiaadghaaeqaaOGaeyOeI0IaamyCaiaadAhacaGGPaGa aiyxaiaac2hadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGSaGaae iiaiaabccacaqGOaGaeqOSdiMaeyypa0ZaaSaaaeaacaqGXaaabaGa ae4samaaBaaaleaacaqGcbaabeaakiaadsfaaaGaaeykaaaa@603E@ This rather unusual expression of Γn occurs because the phonons in the final state are related to a different deformation. However, the analytical evaluation of U(k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaaaa @3C6B@ is a critical step in the calculation of the decay rate Γn. It is well known that the trace contained in U(k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaaaa @3C6B@ can be approximately calculated by using the occupation number states of single-particles and coherent state.

However the former is both a very tedious calculation, including the summation of infinite series, and also not rigorous because the state of the excited quasiparticles is coherent in Pang’s model. Here we use the coherent state to calculate the U(k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaaaa @3C6B@ as it is described in Appendix B. The decay rate obtained finally is Γ n = lim t d W ¯ dt = 2 n μ 1 2 π 2 N 2 k k [ | g 1 (k)+2 g 2 (k) | 2 ( r 0 k) 2 sec h 2 [π(k k ) r 0 /2n μ 1 ] (n μ 1 ) 2 + ( k r 0 ) 2 Re 0 dt { exp[i(nJ ( k r 0 ) 2 +n( n 2 2 3 n) μ 1 2 J t/+ R n (t)+ ξ n (t)] exp[i( ω k kv)t] exp[β( ω k kv)]1 } ]                     (74) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqqHto WrdaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWfqaqaaiGacYgacaGG PbGaaiyBaaWcbaGaamiDaiabgkziUkabg6HiLcqabaGcdaWcaaqaai aadsgaceWGxbGbaebaaeaacaWGKbGaamiDaaaacqGH9aqpdaWcaaqa aGqaaiaa=jdaaeaacaWGUbGaeqiVd02aaSbaaSqaaiaaigdaaeqaaO GaeS4dHG2aaWbaaSqabeaacaWFYaaaaaaakmaalaaabaGaeqiWda3a aWbaaSqabeaacaWFYaaaaaGcbaGaamOtamaaCaaaleqabaGaa8Nmaa aaaaGcdaaeqbqaamaadeaabaGaa8hFaiaadEgadaWgaaWcbaGaa8xm aaqabaGccaWFOaGaam4Aaiaa=LcacqGHRaWkcaWFYaGaam4zamaaBa aaleaacaWFYaaabeaakiaa=HcacaWGRbGaa8xkaiaa=XhadaahaaWc beqaaiaa=jdaaaaakiaawUfaamaalaaabaGaa8hkaiaadkhadaWgaa WcbaGaa8hmaaqabaGccaWGRbGaa8xkamaaCaaaleqabaGaa8Nmaaaa kiGacohacaGGLbGaai4yaiaadIgadaahaaWcbeqaaiaa=jdaaaGcca WFBbGaeqiWdaNaa8hkaiaadUgacqGHsislceWGRbGbauaacaWFPaGa amOCamaaBaaaleaacaWFWaaabeaakiaa=9cacaWFYaGaa8NBaiabeY 7aTnaaBaaaleaacaaIXaaabeaakiaac2faaeaacaGGOaGaamOBaiab eY7aTnaaBaaaleaacaaIXaaabeaakiaacMcadaqhaaWcbaaabaGaaG OmaaaakiabgUcaRiaa=HcaceWGRbGbauaacaWGYbWaaSbaaSqaaiaa =bdaaeqaaOGaa8xkamaaCaaaleqabaGaa8NmaaaaaaaabaGaam4Aai qadUgagaqbaaqab0GaeyyeIuoakiGackfacaGGLbWaa8qmaeaacaWG KbGaamiDaaWcbaGaa8hiaiaa=bdaaeaacqGHEisPa0Gaey4kIipaki abgwSixdqaamaadiaabaWaaiWaaeaaciGGLbGaaiiEaiaacchacaWF BbGaeyOeI0IaamyAaiaa=HcacaWFUbGaamOsaiaa=HcaceWGRbGbau aacaWGYbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkamaaCaaaleqabaGa a8NmaaaakiabgUcaRiaad6gacaGGOaGaamOBamaaCaaaleqabaGaaG OmaaaakiabgkHiTmaalaaabaGaa8Nmaaqaaiaa=ndaaaGaamOBaiaa cMcacqaH8oqBdaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWFkbGaa8 hiaiaadshacaWFVaGaeS4dHGMaey4kaSIaamOuamaaBaaaleaacaWG Ubaabeaakiaa=HcacaWG0bGaa8xkaiabgUcaRiabe67a4naaBaaale aacaWGUbaabeaakiaa=HcacaWG0bGaa8xkaiaa=1fadaWcaaqaaiGa cwgacaGG4bGaaiiCaiaa=TfacaWGPbGaa8hkaiabeM8a3naaBaaale aacaWGRbaabeaakiabgkHiTiaadUgacaWG2bGaa8xkaiaadshacaWF DbaabaGaciyzaiaacIhacaGGWbGaa83waiabek7aIjabl+qiOjaa=H cacqaHjpWDdaWgaaWcbaGaam4AaaqabaGccqGHsislcaWGRbGaamOD aiaa=LcacaWFDbGaeyOeI0Iaa8xmaaaaaiaawUhacaGL9baaaiaaw2 faaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaein aiaabMcaaaaa@EBC2@ Where R n (t)= 1 n 2 N k | α k | 2 {iexp[i( ω k kv)t]} ,     ξ n (t)= 4 n 2 N k | α k | 2 sin 2 [ 1 2 ( ω k kv)t] exp[β( ω k kv)]1        (75) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGUbaabeaakiaacIcacaWG0bGaaiykaiabg2da9iabgkHi TmaalaaabaGaaGymaaqaaiaad6gadaahaaWcbeqaaiaaikdaaaGcca WGobaaamaaqafabaGaaiiFaiabeg7aHnaaBaaaleaacaWGRbaabeaa kiaacYhadaahaaWcbeqaaiaaikdaaaGccaGG7bGaamyAaiabgkHiTi GacwgacaGG4bGaaiiCaiaacUfacqGHsislcaWGPbGaaiikaiabeM8a 3naaBaaaleaacaWGRbaabeaakiabgkHiTiaadUgacaWG2bGaaiykai aadshacaGGDbGaaiyFaaWcbaGaam4Aaaqab0GaeyyeIuoakiaacYca caqGGaGaaeiiaiaabccacaqGGaGaeqOVdG3aaSbaaSqaaiaad6gaae qaaOGaaiikaiaadshacaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacaaI 0aaabaGaamOBamaaCaaaleqabaGaaGOmaaaakiaad6eaaaWaaabuae aadaWcaaqaaiaacYhacqaHXoqydaWgaaWcbaGaam4AaaqabaGccaGG 8bWaaWbaaSqabeaacaaIYaaaaOGaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIYaaaaOGaai4wamaalaaabaGaaGymaaqaaiaaikdaaaGa aiikaiabeM8a3naaBaaaleaacaWGRbaabeaakiabgkHiTiaadUgaca WG2bGaaiykaiaadshacaGGDbaabaGaciyzaiaacIhacaGGWbGaai4w aiabek7aIjabl+qiOjaacIcacqaHjpWDdaWgaaWcbaGaam4Aaaqaba GccqGHsislcaWGRbGaamODaiaacMcacaGGDbGaeyOeI0IaaGymaaaa aSqaaiaadUgaaeqaniabggHiLdGccaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeikaiaabEdacaqG1aGaaeykaaaa@9878@ This is just a generally analytical expression for the decay rate of the soliton containing n quanta at any temperature within lowest order perturbation theory. Note that in the case where a phonon with wave vector k in Eq.(75) is absorbed, the delocalized excitation produced does not need to have wave vector equal to k. The wave vector here is only approximately conserved by the sech2 [π( kk ) r 0 /2n μ 1 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waabaaa aaaaaapeGaeqiWda3damaabmaabaWdbiaadUgacqGHsislcaWGRbGa aiygGaWdaiaawIcacaGLPaaacaWGYbWaaSbaaSqaaiaaicdaaeqaaO Gaai4laiaaikdacaWGUbGaeqiVd02aaSbaaSqaaiaaigdaaeqaaOGa aiyxaaaa@45C3@ term. This is, of course, a consequence of the breaking of the translation symmetry by the deformation. Consequently, we do not find the usual energy conservation. The terms, Rn(t) and ξ n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaa@3B33@ , occur because the phonons in the initial and final states are defined relative to different deformations[24-26].

We should point out that the approximations made in the above calculation are physically justified because the transition and decay of the soliton is mainly determined by the energy of the thermal phonons absorbed. Thus the phonons with large wave vectors, which fulfill wave vector conservation, make a major contribution to the transition matrix element, while the contributions of the phonons with small wave vector, which do not fulfill wave vector conservation, are very small, and can be neglected.

From Eqs.(74) and (75) we see that the Γn and Rn(t) and ξ n (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaaaaa@3B33@ and μ=n μ 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0JaamOBaiabeY7aTnaaBaaaleaacaaIXaaabeaaaaa@3C41@ mentioned above are all changed by increasing the number of quanta, n. Therefore, the approximation methods used to calculate Γn and related quantities (especially the integral contained in Γn) should be different for different n. We now calculate the explicit formula of the decay rate of the new soliton with two-quanta (n=2) by using Eqs. (74)-(75) in Pang’s model. In such a case we can compute explicitly the expressions of this integral and R2 (t) and ξ 2 (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFC@ contained in Eqs.(74)-(75) by means of approximation. As a matter of fact, in Eq. 75) at n=2 the functions R2 (t) and ξ 2 (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFC@ can be exactly evaluated in terms of the digamma function and its derivative. In the case when the soliton velocity approaches zero and the phonon frequency q ω is approximated by w/M MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WG3bacbaGaa83laiaad2eaaSqabaaaaa@3897@ |q|r0, as it is shown in Appendix C. For t MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk ziUkabg6HiLcaa@3A4C@ (because we are interested in the long-time steady behavior) the asymptotic forms of R2(t) and ξ 2 (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFC@ are R 2 (t)= R 0 [ln( 1 2 ω α t)+1.578+ 1 2 iπ]        (76) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIYaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iabgkHi TiaadkfadaWgaaWcbaGaaGimaaqabaGccaGGBbGaciiBaiaac6gaca GGOaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHjpWDdaWgaaWcbaGa eqySdegabeaakiaadshacaGGPaGaey4kaSIaaGymaiaac6cacaaI1a GaaG4naiaaiIdacqGHRaWkdaWcaaqaaGqaaiaa=fdaaeaacaWFYaaa aiaadMgacqaHapaCcaGGDbGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGOaGaae4naiaabAdacaqGPaaaaa@5A67@ ξ 2 (t)π R 0 k B Tt/ (where coth 1 2 ω α t~1)         (77) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaGqaaOGaa8hkaiaadshacaWFPaGaeyisISRa eyOeI0IaeqiWdaNaamOuamaaBaaaleaacaWFWaaabeaakiaadUgada WgaaWcbaGaamOqaaqabaGccaWGubGaamiDaiaa=9cacqWIpecAcaWF GaGaa8hiaiaa=HcacaWF3bGaa8hAaiaa=vgacaWFYbGaa8xzaiaa=b cacaWFGaGaa83yaiaa=9gacaWF0bGaa8hAamaalaaabaGaa8xmaaqa aiaa=jdaaaGaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccaWG0bGaa8 NFaiaa=fdacaWFPaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeikaiaabEdacaqG3aGaaeykaaaa@6320@ i.e.,  Lim t ξ 2 (t)=ηt , η=π R 0 /β=π R 0 k B T/ (78) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGPbGaaeOlaiaabwgacaqGUaGaaeilaiaabckapaWaaCbeaeaa caWGmbGaamyAaiaad2gaaSqaaiaadshacqGHsgIRcqGHEisPaeqaaO GaeqOVdG3aaSbaaSqaaiaaikdaaeqaaGqaaOGaa8hkaiaadshacaWF PaGaeyypa0JaeyOeI0Iaeq4TdGMaamiDaiaa=bcacaWFSaGaa8hiai aa=bcacqaH3oaAcqGH9aqpcqaHapaCcaWGsbWaaSbaaSqaaiaa=bda aeqaaOGaa83laiabek7aIjabl+qiOjabg2da9iabec8aWjaadkfada WgaaWcbaGaa8hmaaqabaGccaWGRbWaaSbaaSqaaiaadkeaaeqaaOGa amivaiaa=9cacqWIpecAcaWFGaGaa8hiaiaa=bcacaWFGaGaa8hiai aa=bcacaWFGaGaa8hiaiaa=bcacaWFOaGaa83naiaa=HdacaWFPaaa aa@69DD@ where R 0 = 4 ( χ 1 + χ 2 ) 2 πw (M/w) 1/2 = 2J μ p r 0 π v 0 ,     ω α = 2 μ p π ( w M ) 1/2 ,   T 0 = ω α / K B   (79) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIWaaabeaakiabg2da9maalaaabaGaaGinaiaacIcacqaH hpWydaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHhpWydaWgaaWcba GaaGOmaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaGcbaGaeqiW daNaeS4dHGMaam4DaaaacaGGOaGaamytaiaac+cacaWG3bGaaiykam aaCaaaleqabaGaaGymaiaac+cacaaIYaaaaOGaeyypa0ZaaSaaaeaa caaIYaGaamOsaiabeY7aTnaaBaaaleaacaWGWbaabeaakiaadkhada WgaaWcbaGaaGimaaqabaaakeaacqaHapaCcqWIpecAcaWG2bWaaSba aSqaaiaaicdaaeqaaaaakiaacYcacaqGGaGaaeiiaiaabccacaqGGa GaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccqGH9aqpdaWcaaqaaiaa ikdacqaH8oqBdaWgaaWcbaGaamiCaaqabaaakeaacqaHapaCaaGaai ikamaalaaabaGaam4Daaqaaiaad2eaaaGaaiykamaaCaaaleqabaGa aGymaiaac+cacaaIYaaaaOGaaiilaiaabccacaqGGaGaaeivamaaBa aaleaacaqGWaaabeaakiabg2da9iabl+qiOjabeM8a3naaBaaaleaa cqaHXoqyaeqaaOGaai4laiaadUeadaWgaaWcbaGaamOqaaqabaGcca qGGaGaaeiiaiaabIcacaqG3aGaaeyoaiaabMcaaaa@7B66@ At R0< 1 and T0< T and R0T/T0< 1 for the protein molecules, one can evaluate the integral including in Eq.(74) by using the approximation which is shown in Appendix C. The result is 1 π Re 0 dt exp{ i[2J (k' r 0 ) 2 + 4 3 J μ p 2 ω k ]t/+ R 2 (t)+ ξ 2 (t) } 1 π (2.43 ω α ) R 0 Γ(1 R 0 ) [ η 2 + (δ(k, k )/) 2 ] (1 R 0 )/2 [ 1 1 2 [ π R 0 2 +(1 R 0 )( δ(k, k ) η ) ] 2 ]      (80) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaadaWcaa qaaGqaaiaa=fdaaeaacqaHapaCcqWIpecAaaGaciOuaiaacwgadaWd XaqaaiaadsgacaWG0baaleaacaWFWaaabaGaeyOhIukaniabgUIiYd GcciGGLbGaaiiEaiaacchadaGadaqaaiabgkHiTiaadMgacaWFBbGa a8NmaiaadQeacaWFOaGaam4AaiaacEcacaWGYbWaaSbaaSqaaiaa=b daaeqaaOGaa8xkamaaCaaaleqabaGaa8NmaaaakiabgUcaRmaalaaa baGaa8hnaaqaaiaa=ndaaaGaamOsaiabeY7aTnaaDaaaleaacaWGWb aabaGaaGOmaaaakiabgkHiTiabl+qiOjabeM8a3naaBaaaleaacaWG Rbaabeaakiaa=1facaWG0bGaa83laiabl+qiOjabgUcaRiaadkfada WgaaWcbaGaaGOmaaqabaGccaWFOaGaamiDaiaa=LcacqGHRaWkcqaH +oaEdaWgaaWcbaGaaGOmaaqabaGccaWFOaGaamiDaiaa=LcaaiaawU hacaGL9baaaeaacqGHijYUdaWcaaqaaiaa=fdaaeaacqaHapaCcqWI pecAaaGaa8hkaiaa=jdacaWFUaGaa8hnaiaa=ndacqaHjpWDdaWgaa WcbaGaeqySdegabeaakiaa=LcadaahaaWcbeqaaiabgkHiTiaadkfa daWgaaadbaGaa8hmaaqabaaaaOGaeu4KdCKaa8hkaiaa=fdacqGHsi slcaWGsbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkaiaa=TfacqaH3oaA daahaaWcbeqaaiaa=jdaaaGccqGHRaWkcaWFOaGaeqiTdqMaa8hkai aadUgacaWFSaGabm4AayaafaGaa8xkaiaa=9cacqWIpecAcaWFPaWa aWbaaSqabeaacaWFYaaaaOGaa8xxamaaCaaaleqabaGaeyOeI0Iaa8 hkaiaa=fdacqGHsislcaWGsbWaaSbaaWqaaiaa=bdaaeqaaSGaa8xk aiaa=9cacaWFYaaaaOWaamWaaeaacaWFXaGaeyOeI0YaaSaaaeaaca WFXaaabaGaa8NmaaaadaWadaqaamaalaaabaGaeqiWdaNaamOuamaa BaaaleaacaWFWaaabeaaaOqaaiaa=jdaaaGaey4kaSIaa8hkaiaa=f dacqGHsislcaWGsbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkamaabmaa baWaaSaaaeaacqaH0oazcaWFOaGaam4Aaiaa=XcaceWGRbGbauaaca WFPaaabaGaeq4TdGMaeS4dHGgaaaGaayjkaiaawMcaaaGaay5waiaa w2faamaaCaaaleqabaGaa8NmaaaaaOGaay5waiaaw2faaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaacIcacaaI4aGaaGimaiaa cMcaaaaa@B972@ where δ(k, k )=2J ( k r 0 ) 2 + 4 3 μ p 2 J ω k , Φ 1 = R 0 π 2 , Φ 2 =[(1 R 0 ) tan 1 ( δ(k, k ) η )]     (81) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaai ikaiaadUgacaGGSaGabm4AayaafaGaaiykaiabg2da9iaaikdacaWG kbacbaGaa8hkaiqadUgagaqbaiaadkhadaWgaaWcbaGaaGimaaqaba GccaWFPaWaaWbaaSqabeaacaWFYaaaaOGaey4kaSYaaSaaaeaacaaI 0aaabaGaaG4maaaacqaH8oqBdaqhaaWcbaGaamiCaaqaaiaaikdaaa GccaWGkbGaeyOeI0IaeS4dHGMaeqyYdC3aaSbaaSqaaiaadUgaaeqa aOGaaiilaiabfA6agnaaBaaaleaacaaIXaaabeaakiabg2da9maala aabaGaamOuamaaBaaaleaacaaIWaaabeaakiabec8aWbqaaiaaikda aaGaaiilaiabfA6agnaaBaaaleaacaWFYaaabeaakiabg2da9iaacU facaWFOaGaa8xmaiabgkHiTiaadkfadaWgaaWcbaGaa8hmaaqabaGc caWFPaGaciiDaiaacggacaGGUbWaaWbaaSqabeaacqGHsislcaWFXa aaaOWaaeWaaeaadaWcaaqaaiabes7aKjaa=HcacaWGRbGaa8hlaiqa dUgagaqbaiaa=LcaaeaacqaH3oaAcqWIpecAaaaacaGLOaGaayzkaa GaaiyxaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabIda caqGXaGaaeykaaaa@76C1@ The decay rate of the soliton, in such an approximation, can be represented, from Eqs.(74) and (80), by Γ 2 = lim t d W ¯ dt = 2 μ p ( π N ) 2 k k [ (k r 0 ) 2 | g 1 (k)+2 g 2 (k) | 2 sec h 2 [(π r 0 /2 μ p )(k k )] [ μ p 2 + ( k r 0 ) 2 ][exp(β ω k )1] (2.43 ω α ) - R 0 { ( η 2 + 1 2 [ 4 3 μ p 2 J+2 ( k r 0 ) 2 J ω k ] 2 ) (1+ R 0 )/2 2 η 2 + [ 4 3 μ p 2 J+2 ( k r 0 ) 2 J ω k ] 2 }{ 1 1 2 [ R 0 π 2 +(1 R 0 )[ 4 3 μ p 2 J+2 ( k r 0 ) 2 J ω k η ] ] 2 } (82) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHto WrdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWfqaqaaiGacYgacaGG PbGaaiyBaaWcbaGaamiDaiabgkziUkabg6HiLcqabaGcdaWcaaqaai aadsgaceWGxbGbaebaaeaacaWGKbGaamiDaaaacqGH9aqpdaWcaaqa aGqaaiaa=jdaaeaacqaH8oqBdaWgaaWcbaGaamiCaaqabaaaaOWaae WaaeaadaWcaaqaaiabec8aWbqaaiaad6eaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaWFYaaaaOWaaabuaeaadaWabaqaamaalaaabaGaa8 hkaiaadUgacaWGYbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkamaaCaaa leqabaGaa8Nmaaaakiaa=XhacaWGNbWaaSbaaSqaaiaa=fdaaeqaaO Gaa8hkaiaadUgacaWFPaGaey4kaSIaa8NmaiaadEgadaWgaaWcbaGa a8NmaaqabaGccaWFOaGaam4Aaiaa=LcacaWF8bWaaWbaaSqabeaaca WFYaaaaOGaci4CaiaacwgacaGGJbGaamiAamaaCaaaleqabaGaa8Nm aaaakiaa=TfacaWFOaGaeqiWdaNaamOCamaaBaaaleaacaWFWaaabe aakiaa=9cacaWFYaGaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaa8xk aiaa=HcacaWGRbGaeyOeI0Iabm4AayaafaGaa8xkaiaa=1faaeaaca GGBbGaeqiVd02aa0baaSqaaiaadchaaeaacaaIYaaaaOGaey4kaSIa a8hkaiqadUgagaqbaiaadkhadaWgaaWcbaGaa8hmaaqabaGccaWFPa WaaWbaaSqabeaacaWFYaaaaOGaa8xxaiaa=TfaciGGLbGaaiiEaiaa cchacaWFOaGaeqOSdiMaeS4dHGMaeqyYdC3aaSbaaSqaaiaadUgaae qaaOGaa8xkaiabgkHiTiaa=fdacaWFDbaaaiaa=HcacaWFYaGaa8Nl aiaa=rdacaWFZaGaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccaWFPa WaaWbaaSqabeaacaWFTaGaa8NuamaaBaaameaacaWFWaaabeaaaaaa kiaawUfaaaWcbaGaam4AaiqadUgagaqbaaqab0GaeyyeIuoaaOqaam aacmaabaWaaSaaaeaadaqadaqaaiabeE7aOnaaCaaaleqabaGaa8Nm aaaakiabgUcaRmaalaaabaGaa8xmaaqaaiabl+qiOnaaCaaaleqaba Gaa8NmaaaaaaGccaWFBbWaaSaaaeaacaWF0aaabaGaa83maaaacqaH 8oqBdaqhaaWcbaGaamiCaaqaaiaaikdaaaGccaWGkbGaey4kaSIaa8 Nmaiaa=HcaceWGRbGbauaacaWGYbWaaSbaaSqaaiaa=bdaaeqaaOGa a8xkamaaCaaaleqabaGaa8NmaaaakiaadQeacqGHsislcqWIpecAcq aHjpWDdaWgaaWcbaGaam4AaaqabaGccaWFDbWaaWbaaSqabeaacaWF YaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWFOaGaa8xmaiabgU caRiaadkfadaWgaaadbaGaa8hmaaqabaWccaWFPaGaa83laiaa=jda aaaakeaacqWIpecAdaahaaWcbeqaaiaa=jdaaaGccqaH3oaAdaahaa Wcbeqaaiaa=jdaaaGccqGHRaWkcaWFBbWaaSaaaeaacaWF0aaabaGa a83maaaacqaH8oqBdaqhaaWcbaGaamiCaaqaaiaaikdaaaGccaWGkb Gaey4kaSIaa8Nmaiaa=HcaceWGRbGbauaacaWGYbWaaSbaaSqaaiaa =bdaaeqaaOGaa8xkamaaCaaaleqabaGaa8NmaaaakiaadQeacqGHsi slcqWIpecAcqaHjpWDdaWgaaWcbaGaam4AaaqabaGccaWFDbWaaWba aSqabeaacaWFYaaaaaaaaOGaay5Eaiaaw2haamaacmaabaGaa8xmai abgkHiTmaalaaabaGaa8xmaaqaaiaa=jdaaaWaamWaaeaadaWcaaqa aiaadkfadaWgaaWcbaGaa8hmaaqabaGccqaHapaCaeaacaWFYaaaai abgUcaRiaa=HcacaWFXaGaeyOeI0IaamOuamaaBaaaleaacaWFWaaa beaakiaa=LcadaWadaqaamaalaaabaWaaSaaaeaacaWF0aaabaGaa8 3maaaacqaH8oqBdaqhaaWcbaGaamiCaaqaaiaaikdaaaGccaWGkbGa ey4kaSIaa8Nmaiaa=HcaceWGRbGbauaacaWGYbWaaSbaaSqaaiaa=b daaeqaaOGaa8xkamaaCaaaleqabaGaa8NmaaaakiaadQeacqGHsisl cqWIpecAcqaHjpWDdaWgaaWcbaGaam4AaaqabaaakeaacqWIpecAcq aH3oaAaaaacaGLBbGaayzxaaaacaGLBbGaayzxaaWaaWbaaSqabeaa caWFYaaaaaGccaGL7bGaayzFaaGaaeiiaiaabIcacaqG4aGaaeOmai aabMcaaaaa@0873@ This is the final analytical expression for the decay rate of the quasi-coherent solition with two-quanta. Evidently, it is different from that in the Davydov model [15,21]. To emphasis the difference of the decay rate between the two models we rewrite down the corresponding quantity for the Davydov soliton [15,21] Γ D = 1 2 χ 1 2 μ D ( 2π N ) 2 k k ( 2M ω k ) (k r 0 ) 2 si n 2 (k r 0 )sec h 2 [(π r 0 /2 μ D )(k k )] [ μ D 2 + ( k r 0 ) 2 ][exp(β ω k )1] ( ω α D η D ) R 0 D      (83)          2 η D 2 η D 2 +[J μ D 2 /3+J ( k ' r 0 ) 2 ω k ]                                                                  MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHto WrdaWgaaWcbaGaamiraaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa cqWIpecAdaahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaacqaHhpWyda qhaaWcbaGaaGymaaqaaiaaikdaaaaakeaacqaH8oqBdaWgaaWcbaGa amiraaqabaaaaOWaaeWaaeaadaWcaaqaaiaaikdacqaHapaCaeaaca WGobaaaaGaayjkaiaawMcaamaaCaaaleqabaacbaGaa8Nmaaaakmaa qafabaWaaeWaaeaadaWcaaqaaiabl+qiObqaaiaaikdacaWGnbGaeq yYdC3aaSbaaSqaaiaadUgaaeqaaaaaaOGaayjkaiaawMcaaaWcbaGa am4AaiqadUgagaqbaaqab0GaeyyeIuoakmaalaaabaGaa8hkaiaadU gacaWGYbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkamaaCaaaleqabaGa a8Nmaaaakiaa=nhacaWFPbGaa8NBamaaCaaaleqabaGaa8Nmaaaaki aa=HcacaWGRbGaamOCamaaBaaaleaacaaIWaaabeaakiaa=LcaciGG ZbGaaiyzaiaacogacaWGObWaaWbaaSqabeaacaWFYaaaaOGaa83wai aa=HcacqaHapaCcaWGYbWaaSbaaSqaaiaa=bdaaeqaaOGaa83laiaa =jdacqaH8oqBdaWgaaWcbaGaamiraaqabaGccaWFPaGaa8hkaiaadU gacqGHsislceWGRbGbauaacaWFPaGaa8xxaaqaaiaacUfacqaH8oqB daqhaaWcbaGaamiraaqaaiaaikdaaaGccqGHRaWkcaWFOaGabm4Aay aafaGaamOCamaaBaaaleaacaWFWaaabeaakiaa=LcadaahaaWcbeqa aiaa=jdaaaGccaWFDbGaa83waiGacwgacaGG4bGaaiiCaiaa=Hcacq aHYoGycqWIpecAcqaHjpWDdaWgaaWcbaGaam4AaaqabaGccaWFPaGa eyOeI0Iaa8xmaiaa=1faaaWaaeWaaeaadaWcaaqaaiabeM8a3naaDa aaleaacqaHXoqyaeaacaWGebaaaaGcbaGaeq4TdG2aaSbaaSqaaiaa dseaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0Iaam OuamaaDaaameaacaaIWaaabaGaamiraaaaaaqcLbEacqGHflY1kiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabIdacaqGZaGaae ykaaqaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaWaaSaaaeaacqWIpecAdaahaaWcbeqaaiaaikdaaaGccqaH3o aAdaWgaaWcbaGaamiraaqabaaakeaacqWIpecAdaahaaWcbeqaaiaa ikdaaaGccqaH3oaAdaqhaaWcbaGaamiraaqaaiaaikdaaaGccqGHRa WkcaGGBbGaamOsaiabeY7aTnaaDaaaleaacaWGebaabaGaaGOmaaaa kiaac+cacaaIZaGaey4kaSIaamOsaiaacIcacaWGRbWaaWbaaSqabe aacaGGNaaaaOGaamOCamaaBaaaleaacaaIWaaabeaakiaacMcadaah aaWcbeqaaiaaikdaaaGccqGHsislcqWIpecAcqaHjpWDdaWgaaWcba Gaam4AaaqabaGccaGGDbaaaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccaaaaa@EFD3@ where η D =π R 0 D K B T/, R 0 D = 2 χ 1 2 πw ( M w ) 1/2 , ω α D = 2 μ D π ( M w ) 1/2     (84) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaiaadseaaeqaaOGaeyypa0JaeqiWdaNaamOuamaaDaaaleaa caaIWaaabaGaamiraaaakiaadUeadaWgaaWcbaGaamOqaaqabaGcca WGubGaai4laiabl+qiOjaacYcacaWGsbWaa0baaSqaaiaaicdaaeaa caWGebaaaOGaeyypa0ZaaSaaaeaacaaIYaGaeq4Xdm2aa0baaSqaai aaigdaaeaacaaIYaaaaaGcbaGaeqiWdaNaeS4dHGMaam4DaaaacaGG OaWaaSaaaeaacaWGnbaabaGaam4DaaaacaGGPaWaaWbaaSqabeaaca aIXaGaai4laiaaikdaaaGccaGGSaGaeqyYdC3aa0baaSqaaiabeg7a HbqaaiaadseaaaGccqGH9aqpdaWcaaqaaiaaikdacqaH8oqBdaWgaa WcbaGaamiraaqabaaakeaacqaHapaCaaGaaiikamaalaaabaGaamyt aaqaaiaadEhaaaGaaiykamaaCaaaleqabaGaaGymaiaac+cacaaIYa aaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG4aGaaeinaiaa bMcaaaa@6AB5@ Equation (83) can also be found out from Eq.(74) at n=1 by using the Cottingham et al’s approximation.

The two formulae above, Eqs. (82) and (83), are completely different, not only for the parameter’s values, but also the factors contained in them. In Eq.(82) the factor, { 1 1 2 [ R 0 π 2 +(1 R 0 )[ ( 4 3 μ p 2 J+2 ( k ' r 0 ) 2 J ω k )/η ] ] 2 } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaaca aIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaiaawUhaamaa ciaabaWaamWaaeaadaWcaaqaaiaadkfadaWgaaWcbaGaaGimaaqaba GccqaHapaCaeaacaaIYaaaaiabgUcaRiaacIcacaaIXaGaeyOeI0Ia amOuamaaBaaaleaacaaIWaaabeaakiaacMcadaWadaqaaiaacIcada WcaaqaaiaaisdaaeaacaaIZaaaaiabeY7aTnaaDaaaleaacaWGWbaa baGaaGOmaaaakiaadQeacqGHRaWkcaaIYaGaaiikaiaadUgadaahaa WcbeqaaiaacEcaaaGccaWGYbWaaSbaaSqaaiaaicdaaeqaaOGaaiyk amaaCaaaleqabaGaaGOmaaaakiaadQeacqGHsislcqWIpecAcqaHjp WDdaWgaaWcbaGaam4AaaqabaGccaGGPaGaai4laiabl+qiOjabeE7a ObGaay5waiaaw2faaaGaay5waiaaw2faamaaCaaaleqabaGaaGOmaa aaaOGaayzFaaaaaa@6280@ is added, while in Eq.(83) the factor, ( ω α η D ) R 0 D η D MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaala aabaGaeqyYdC3aaSbaaSqaaiabeg7aHbqabaaakeaacqaH3oaAdaWg aaWcbaGaamiraaqabaaaaOGaaiykamaaCaaaleqabaGaeyOeI0Iaam OuamaaDaaameaacaaIWaaabaGaamiraaaaaaGccqaH3oaAdaWgaaWc baGaamiraaqabaaaaa@43F8@ replaces the term (2.43 ω d ) R 0 ( η 2 + 1 2 [ 4 3 μ p 2 J+2 ( k ' r 0 ) 2 J ω k ] 2 ) ( 1+ R 0 2 ) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaik dacaGGUaGaaGinaiaaiodacqaHjpWDdaWgaaWcbaGaamizaaqabaGc caGGPaWaaWbaaSqabeaacqGHsislcaWGsbWaaSbaaWqaaiaaicdaae qaaaaakiabgwSixlaacIcacqaH3oaAdaahaaWcbeqaaiaaikdaaaGc cqGHRaWkdaWcaaqaaiaaigdaaeaacqWIpecAdaahaaWcbeqaaiaaik daaaaaaOGaai4wamaalaaabaGaaGinaaqaaiaaiodaaaGaeqiVd02a a0baaSqaaiaadchaaeaacaaIYaaaaOGaamOsaiabgUcaRiaaikdaca GGOaGaam4AamaaCaaaleqabaGaai4jaaaakiaadkhadaWgaaWcbaGa aGimaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaamOsaiabgk HiTiabl+qiOjabeM8a3naaBaaaleaacaWGRbaabeaakiaac2fadaah aaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabeaacaGGOaWaaSaaae aacaaIXaGaey4kaSIaamOuamaaBaaameaacaaIWaaabeaaaSqaaiaa ikdaaaGaaiykaaaakiaacMcaaaa@66A5@ in Eq.(82) due to the two-quanta nature of the new wave function and the additional interaction term in the new Hamiltonian. In Eq. (82) the η, R0 and T0 are not small, unlike in the Davydov model. Using Eq.(72) and table 1 we find out the values of η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A1@ ,R0 and To at T=300K in both models, which are listed in Table 2. From this table we see that the η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A1@ , Ro and To for Pang’s model are about 3 times larger than the corresponding values in the Davydov model due to the increases of μp and of the non-linear interaction coefficient Gp. Thus the approximations used in the Davydov model by Cottingham, et al. [14] cannot be applied in our calculation of lifetime of the new soliton, although we utilized the same quantum-perturbation scheme. Hence we can audaciously suppose that the lifetimes of the quasi-coherent soliton will greatly change.
Table 2: Comparison of characteristic parameters in the Davydov model and in our new model

 

Ro

To (K)

η(× 10 13 /s) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGMaai ikaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaG4maaaa kiaac+cacaWGZbGaaiykaaaa@3FE0@

New model

0.529

294

6.527

Davydov model

0.16

95

2.096

Discussion for the Lifetime of the New Soliton and Results
The above expression, Eq. (82), allows us to compute numerically the decay rate, , and the lifetimes of the new soliton, τ= 1/ , for values of the physical parameters appropriate to -helical protein molecules. Using the parameter values given in Eq.(37), tables 1 and 2 , v=0.2v0 and assuming the wave vectors are in the Brillouin zone, the values of between 1.54×1010S-1 - 1.89×1010S-1 can be obtained. This corresponds to the soliton lifetimes τ, of between 0.53×10-10S- 0.65×10-10S at T=300K, or τ/ τ0=510-630, where τ/τ0=r0/v0 is the time for travelling one lattice spacing at the speed of sound, equal to (M/w)1/2=0.96×10-13S. In this amount of time, the new soliton, travelling at two tenths of the speed of sound in the chain, would travel several hundreds of amino acid lattice spacings, that is several hundred times more than the Davydov soliton for which τ/τ0< 10 at 300K [15,21] (i.e., the Davydov soliton traveling at a half of the sound speed can cover less than 10 lattice spacing in its lifetime) The lifetime is sufficiently long for the new soliton excitation to be a carrier of bio-energy. Therefore the quasi-coherent soliton is a viable mechanism for the bio-energy transport at biological temperature in the above range of parameters.

Attention is being paid to the relationship between the lifetime of the quasi-coherent soliton and temperature. Fig.3 shows the relative lifetimes τ/τ0 of the new soliton versus temperature T for a set of widely accepted parameter values as shown in Eq.(37). Since one assumes that v< v0, the soliton will not travel the length of the chain unless τ/τ0 is large compared with L/r0, where L=Nr0 is the typical length of the protein molecular chains. Hence for L/r0≈100,τ/τ0>500 is a reasonable criterion for the soliton to be a possible mechanism of the bio-energy transport in protein molecules. The lifetime of the quasi-coherent soliton shown in Figure.4 decreases rapidly as temperature increases, but below T=310K it is still large enough to fulfill the criterion.
Table 3: Comparison of features of the solitons between our model and Davydov model

Model

nonlinear interaction G(10-21J)

Amplitude

Width 10-10m

Binding energy
(10-21J)

Lifetime at 300K
(S)

Critical temperature
(K)

Number of amino acid traveled by soliton in lifetime

Our model

3.8

1.72

4.95

-7.8

10-9-10-10

320

Several handreds

Davydov model

1.18

0.974

14.88

-0.188

10-12-10-13

<200

Less than 10

Thus the new soliton can play an important role in biological processes.

For comparison, log versus the temperature relationships was plotted simultaneously for the Davydov soliton and the new soliton with a quasi-coherent two-quanta state in Figure 4. The temperature-dependence of log ( τ/τ0) of the Davydov soliton is obtained from Eq. (83). We find that the differences of values of ( τ/τ0) between the two models are very large. The value of ( τ/τ0) of the Davydov soliton really is too small, and it can only travel fewer than ten lattice spacings in half the speed of sound in the protein chain. Hence it is true that the Davydov soliton is ineffective for biological processes [3-23].
Figure 3: Soliton lifetime τ relatively to τ0 as a function of the temperatureT for parameters appropriate to the α-helical molecules in Pan’s model in Eq. (9)
Figure 4: log(τ/τ0) versus the temperature. The solid line is the result of Pang’s model, the dashed line is the result of the Davydov model.
The dependency of the soliton lifetime on the other parameters can also be studied by using Eq. (82). Parameter values near the above accepted values shown in Eq. (37) are chosen. In Pang’s model we know from Eq. (82) that the lifetime of the soliton depends mainly on the following parameters: coupling constants (χ12),, M, w, J, phonon energy ω k , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4dHGMaeq yYdC3aaSbaaSqaaiaadUgaaeqaaOGaaiilaaaa@3AC1@ , as well as well as on the composite parameters μ(μ=μp ), R0 and T/T0. At a given temperature, τ/τ0 increases as μ and T0 increase. The dependences of the lifetime τ/τ0 ,at 300K on ((χ12)) and μ are shown in Figs.5 and 6, respectively . Since μ is inversely proportional to the size of the soliton, and determines the binding energy in the new model, it is an important quantity. It is regarded as an independent variable. In such a case the other parameters in Eq. (82) adopt the values in Eq. (37). It is clear from Figs.5 and 6 that the lifetime of the soliton, τ/τ0, increases rapidly with increasing μand ((χ12)). Furthermore, when μ≥5.8 and ((χ12))≥7.5×10-11N, which are values appropriate to the new model, we find τ/τ0>500.

For comparison, the corresponding result obtained using Eq. (83) is shown for the original Davydov model as a dashed line in Fig.6. Here we see that the increase in lifetime of the Davydov soliton with increasing μ is quite slow and the difference between the two models increases rapidly with increasing μ. The same holds for the dependency on the parameter (χ12), but the result for the Davydov soliton is not drawn in Figure5. These results show again that the quasi-coherent soliton in Pang’s model is a likely candidate for the mechanism of bio-energy transport in the protein molecules. In addition it shows that a basic mechanism for increasing the lifetime of the soliton in the biomacromolecules is to enhance the strength of the exciton-phonon interaction.
Figure 5: τ/τ0versus (χ12) relation in Eq. (82)
Figure 6: τ/τ0versus μ relation. The solid and dashed lines are results of Eq. (82) and Eq.(83),respectively
Figure 7: τ/τ0 versus η relation in Eq. (82)
In Figure 7, τ/τ0 versus η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A1@ is plotted. Since η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq 4TdGgaaa@388E@ designates the influence of the thermal phonons on the soliton, it is also an important quantity. Thus, it is regarded here as an independent variable. The other parameters in Eq.(82) take the values in Eq. (37). From this figure we see that τ/τ0 increases with increasing η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A1@ . Therefore, to enhance η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A1@ can also increase the value of τ/τ0.

In order to understand the behavior of the quasi-coherent soliton lifetime in very wide ranges, it is necessary to study τ/τ0 in the limit R 2 ( t ) R 0 [ i π 2 ω α t/6+3ζ( 3 ) ( ω α t ) 2 ]0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWaaeWaaeaapeGa amiDaaWdaiaawIcacaGLPaaapeGaeyisISRaaiifGiaadkfapaWaaS baaSqaa8qacaaIWaaapaqabaGcdaWadaqaa8qacaWGPbGaeqiWda3d amaaCaaaleqabaWdbiaaikdaaaGccqaHjpWDpaWaaSbaaSqaa8qacq aHXoqya8aabeaak8qacaWG0bGaai4laiaaiAdacqGHRaWkcaaIZaGa eqOTdO3damaabmaabaWdbiaaiodaa8aacaGLOaGaayzkaaWaaeWaae aapeGaeqyYdC3damaaBaaaleaapeGaeqySdegapaqabaGcpeGaamiD aaWdaiaawIcacaGLPaaadaahaaWcbeqaa8qacaaIYaaaaaGcpaGaay 5waiaaw2faa8qacqGHsgIRcaaIWaaaaa@5B40@ in Eq.(75) or Eqs.(C1) and (C3) (i.e., this is in the initial case) in which we can evaluate analytically the values of R2(t) and ξ2(t). In fact, for ω a t<1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadggaaeqaaOGaamiDaiabgYda8iaaigdaaaa@3B96@ both R2(t) and ξ2(t) have power-series expansions. To the lowest order as ω a t0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadggaaeqaaOGaamiDaabaaaaaaaaapeGaeyOKH4QaaGim aaaa@3C9E@ , it can be found from Eq.(75) R 2 ( t ) R 0 [ i π 2 ω α t/6+3ζ( 3 ) ( ω α t ) 2 ]     (85) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWaaeWaaeaapeGa amiDaaWdaiaawIcacaGLPaaapeGaeyisISRaaiifGiaadkfapaWaaS baaSqaa8qacaaIWaaapaqabaGcdaWadaqaa8qacaWGPbGaeqiWda3d amaaCaaaleqabaWdbiaaikdaaaGccqaHjpWDpaWaaSbaaSqaa8qacq aHXoqya8aabeaak8qacaWG0bGaai4laiaaiAdacqGHRaWkcaaIZaGa eqOTdO3damaabmaabaWdbiaaiodaa8aacaGLOaGaayzkaaWaaeWaae aapeGaeqyYdC3damaaBaaaleaapeGaeqySdegapaqabaGcpeGaamiD aaWdaiaawIcacaGLPaaadaahaaWcbeqaa8qacaaIYaaaaaGcpaGaay 5waiaaw2faaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaa bIdacaqG1aGaaeykaaaa@5E82@ ξ 2 (t) R 0 K B 2 T T 0 π 2 3 2 t 2 ,           (86) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaGaeyisISRaeyOe I0YaaSaaaeaacaWGsbWaaSbaaSqaaiaaicdaaeqaaOGaam4samaaDa aaleaacaWGcbaabaGaaGOmaaaakiaadsfacaWGubWaaSbaaSqaaiaa icdaaeqaaOGaeqiWda3aaWbaaSqabeaacaaIYaaaaaGcbaGaaG4mai abl+qiOnaaCaaaleqabaGaaGOmaaaaaaGccaWG0bWaaWbaaSqabeaa caaIYaaaaOGaaeilaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabIdacaqG 2aGaaeykaaaa@568D@  using  coth (π ω α t)[ (π ω α t) 1 + π 3 ω α t]      (87) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaqGGa GaaeyDaiaabohacaqGPbGaaeOBaiaabEgacaqGGaGaaeiiaiaaboga caqGVbGaaeiDaiaabIgakiaabccacaqGOaGaeqiWdaNaeqyYdC3aaS baaSqaaiabeg7aHbqabaGccaWG0bGaaiykaiabgIKi7kaacUfacaGG OaGaeqiWdaNaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccaWG0bGaai ykamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabgUcaRmaalaaabaGa eqiWdahabaGaaG4maaaacqaHjpWDdaWgaaWcbaGaeqySdegabeaaki aadshacaGGDbGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaabIdacaqG3aGaaeykaaaa@64FF@ Thus 1 π Re 0 dtexp{ i[ 2J (k' r 0 ) 2 + 4J μ p 2 3 ω k ] t + R 2 (t)+ ξ 2 (t) } [ 4π(3ζ(3) R 0 K B 2 T 0 2             + R 0 π 2 K B 2 T T 0 /3) ] 1 2 exp{ [2J (k' r 0 ) 2 + 4 3 μ p 2 J ω k +( R 0 π 2 K B T)] 2 4[3ζ(3) R 0 K B 2 T 0 2 + R 0 π 2 K B 2 T T 0 /3] }              (88) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaigdaaeaacqaHapaCcqWIpecAaaGaciOuaiaacwgadaWdXaqa aaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaamizaiaadshaci GGLbGaaiiEaiaacchadaGadaqaaiabgkHiTiaadMgadaWadaqaaiaa ikdacaWGkbGaaiikaiaadUgacaGGNaGaamOCamaaBaaaleaacaaIWa aabeaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaWcaaqa aiaaisdacaWGkbGaeqiVd02aa0baaSqaaiaadchaaeaacaaIYaaaaa GcbaGaaG4maaaacqGHsislcqWIpecAcqaHjpWDdaWgaaWcbaGaam4A aaqabaaakiaawUfacaGLDbaadaWccaqaaiaadshaaeaacqWIpecAaa Gaey4kaSIaamOuamaaBaaaleaacaaIYaaabeaakiaacIcacaWG0bGa aiykaiabgUcaRiabe67a4naaBaaaleaacaaIYaaabeaakiaacIcaca WG0bGaaiykaaGaay5Eaiaaw2haaiabgIKi7oaadeaabaGaaGinaiab ec8aWjaacIcacaaIZaGaeqOTdONaaiikaiaaiodacaGGPaGaamOuam aaBaaaleaacaaIWaaabeaakiaadUeadaqhaaWcbaGaamOqaaqaaiaa ikdaaaGccaWGubWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGccaGLBb aaaeaaaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiamaadiaabaGaey4kaSIaamOuam aaBaaaleaacaaIWaaabeaakiabec8aWnaaCaaaleqabaGaaGOmaaaa kiaadUeadaqhaaWcbaGaamOqaaqaaiaaikdaaaGccaWGubGaamivam aaBaaaleaacaaIWaaabeaakiaac+cacaaIZaGaaiykaaGaayzxaaWa aWbaaSqabeaacqGHsisldaWccaqaaiaaigdaaeaacaaIYaaaaaaaki GacwgacaGG4bGaaiiCamaacmaabaGaeyOeI0YaaSaaaeaacaGGBbGa aGOmaiaadQeacaGGOaGaam4AaiaacEcacaWGYbWaaSbaaSqaaiaaic daaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaa baGaaGinaaqaaiaaiodaaaGaeqiVd02aa0baaSqaaiaadchaaeaaca aIYaaaaOGaamOsaiabgkHiTiabl+qiOjabeM8a3naaBaaaleaacaWG RbaabeaakiabgUcaRiabl+qiOjaacIcacaWGsbWaaSbaaSqaaiaaic daaeqaaOGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGaam4samaaBaaa leaacaWGcbaabeaakiaadsfacaGGPaGaaiyxamaaCaaaleqabaGaaG OmaaaaaOqaaiaaisdacaGGBbGaaG4maiabeA7a6jaacIcacaaIZaGa aiykaiaadkfadaWgaaWcbaGaaGimaaqabaGccaWGlbWaa0baaSqaai aadkeaaeaacaaIYaaaaOGaamivamaaDaaaleaacaaIWaaabaGaaGOm aaaakiabgUcaRiaadkfadaWgaaWcbaGaaGimaaqabaGccqaHapaCda ahaaWcbeqaaiaaikdaaaGccaWGlbWaa0baaSqaaiaadkeaaeaacaaI YaaaaOGaamivaiaadsfadaWgaaWcbaGaaGimaaqabaGccaGGVaGaaG 4maiaac2faaaaacaGL7bGaayzFaaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGOaGaaeioaiaabEdacaqGPaaaaaa@DF6F@ when T/T0>1 and π40T/2μT0>1. The above integral is a generalization of the usual δ - function for energy conservation in zero-temperature perturbation theory. Thus we can obtain from Eqs.(74) and (87) at n=2 the decay rate of the soliton as Γ 2 = 2 π 3 μ p N 2 K B ( π R 0 T 0 [3ζ(3) T 0 + π 2 T/3] ) 1 2 k k (k r 0 ) 2 | g 1 (k)+2 g 2 (k) | 2 μ p 2 + ( k r 0 ) 2 sec h 2 [ ( π r 0 2 μ p )(k k ) ]       { exp[ [2J ( k r 0 ) 2 + 4 3 μ p 2 J ω k + 1 6 R 0 π 2 K B T 0 ] 2 4[3ζ(3) R 0 K B 2 T 0 2 + R 0 K B 2 T T 0 π 2 /3] ][ exp(β ω k )1 ] } 1                                  MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHto WrdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaaikdacqaH apaCdaahaaWcbeqaaiaaiodaaaaakeaacqaH8oqBdaWgaaWcbaGaam iCaaqabaGccqWIpecAcaWGobWaaWbaaSqabeaacaaIYaaaaOGaam4s amaaBaaaleaacaWGcbaabeaaaaGcdaqadaqaamaalaaabaGaeqiWda habaGaamOuamaaBaaaleaacaaIWaaabeaakiaadsfadaWgaaWcbaGa aGimaaqabaGccaGGBbGaaG4maiabeA7a6jaacIcacaaIZaGaaiykai aadsfadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHapaCdaahaaWc beqaaiaaikdaaaGccaWGubGaai4laiaaiodacaGGDbaaaaGaayjkai aawMcaamaaCaaaleqabaGaeyOeI0YaaSGaaeaacaaIXaaabaGaaGOm aaaaaaGcdaaeqbqaamaalaaabaGaaiikaiaadUgacaWGYbWaaSbaaS qaaiaaicdaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiaacYha caWGNbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadUgacaGGPaGaey 4kaSIaaGOmaiaadEgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaam4A aiaacMcacaGG8bWaaWbaaSqabeaacaaIYaaaaaGcbaGaeqiVd02aa0 baaSqaaiaadchaaeaacaaIYaaaaOGaey4kaSIaaiikaiqadUgagaqb aiaadkhadaWgaaWcbaGaaGimaaqabaGccaGGPaWaaWbaaSqabeaaca aIYaaaaaaakiGacohacaGGLbGaai4yaiaadIgadaahaaWcbeqaaiaa ikdaaaGcdaWadaqaamaabmaabaWaaSaaaeaacqaHapaCcaWGYbWaaS baaSqaaiaaicdaaeqaaaGcbaGaaGOmaiabeY7aTnaaBaaaleaacaWG WbaabeaaaaaakiaawIcacaGLPaaacaGGOaGaam4AaiabgkHiTiqadU gagaqbaiaacMcaaiaawUfacaGLDbaaaSqaaiaadUgaceWGRbGbauaa aeqaniabggHiLdaakeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiam aacmaabaGaciyzaiaacIhacaGGWbWaamWaaeaadaWcaaqaaiaabUfa caaIYaGaamOsaiaabIcaceWGRbGbauaacaWGYbWaaSbaaSqaaiaaic daaeqaaOGaaeykamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaa baGaaGinaaqaaiaaiodaaaGaamiVdmaaDaaaleaacaWGWbaabaGaaG OmaaaakiaadQeacqGHsislcqWIpecAcqaHjpWDdaWgaaWcbaGaam4A aaqabaGccqGHRaWkdaWcaaqaaiaabgdaaeaacaqG2aaaaiaabkfada WgaaWcbaGaaeimaaqabaGccqaHapaCdaahaaWcbeqaaiaabkdaaaGc caqGlbWaaSbaaSqaaiaabkeaaeqaaOGaaeivamaaBaaaleaacaqGWa aabeaakiaab2fadaahaaWcbeqaaiaabkdaaaaakeaacaaI0aGaai4w aiaaiodacqaH2oGEcaGGOaGaaG4maiaacMcacaWGsbWaaSbaaSqaai aaicdaaeqaaOGaam4samaaDaaaleaacaWGcbaabaGaaGOmaaaakiaa dsfadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccqGHRaWkcaWGsbWaaS baaSqaaiaaicdaaeqaaOGaam4samaaDaaaleaacaWGcbaabaGaaGOm aaaakiaadsfacaWGubWaaSbaaSqaaiaaicdaaeqaaOGaeqiWda3aaW baaSqabeaacaaIYaaaaOGaai4laiaaiodacaGGDbaaaaGaay5waiaa w2faamaadmaabaGaciyzaiaacIhacaGGWbGaaiikaiabek7aIjabl+ qiOjabeM8a3naaBaaaleaacaWGRbaabeaakiaacMcacqGHsislcaaI XaaacaGLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaSqabeaacqGHsi slcaaIXaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccaaaaa@F680@
Figure 8: τ/τ0vs T relation in Pang”s model in Eq. (88)
Figure 9: τ/τ0versus (χ12), relations in the new model in Eq. (88)
Figure 10: τ/τ0 versus μ relation in the new model in Eq. (88)
Figure 11: τ/τ0 versus T0 relation .Here the solid and dashed lines are the results in the new model in Eq.(88) and in the Davydov model, respectively
In Figure9 and 10 we plot τ/τ0 versus (χ12) and versus μ at T=300K, respectively. From Figure 8-10 it can be seen that τ/τ0 increases as T decreases and as μ and (χ12) increase. Furthermore, it is clear from this Gaussian expression in Eq.(88) that the lifetime of the new soliton will be large if μ and (χ12) are larger, but the Gaussian expression is very small for k and k’ between -π/r0 and +π/r0, i.e., in the Brillouin zero. Obviously, the temperature dependence of the lifetime of the new soliton is mainly due to the temperature dependence of the width of the Gaussian, which decreases with decreasing temperature. The dashed line in Fig.10 is the result for the Davydov soliton under the same conditions. It is clear that the lifetime of the Davydov soliton is lower than that of the new soliton, especially at large , although at low the difference between them is small. Taking Figure 4 also into account we find that the lifetime of the Davydov soliton is indeed generally low. However this is not the case for the new soliton. In Figure 11, τ/τ0 is plotted as a function of T0 at T=300K. T0 is related to the Debye temperature of the systems, therefore it is also an important quantity. It is regarded here as an independent variable and evaluate other parameters as in Eq. (37). From this figure it can seen that the lifetime of the new soliton is large if T0 is either large or small, because the Gaussian expression in Eq.(88) is very small for k and k’ between -π/r0 and +π/r0. As a point of reference, note that these parameters have the values τ/τ0≈1.03 —1.06, JT/KBT =4.10 at 300K and μ=5.81- 5.96 depending on whether the widely accepted or the“threechannel” parameter values for the protein are assumed. From these results, it is clear that using widely accepted and realistic parameter values, the new model can satisfy the relation τ/ τ00≥500 at 300K and large μ and large T0. Hence the proposed new soliton model provides a viable candidate for the biological processes.
Conclusions
Here a new theory of bio-energy transport is proposed to study the properties of the nonlinear excitation and motion of the soliton along protein molecules. In this theory, Davydov’s Hamiltonian and wave function of the systems are simultaneously improved and extended, a new interaction is added into the original Hamiltonian, and the original wave function of the excitation state of single particles is replaced by a new wave function of a two-quanta quasi-coherent state. From this model, a lot of interesting and new results are obtained. The soliton has sufficiently long lifetime and can pay an important role in biological processes. Therefore, it is an exact carrier of bio-energy in living systems. Present problem is why the quasi-coherent soliton has such long lifetime? From Eqs. (35) and (45) and tables 1 and 2 it can be seen that the binding energy and localization of the new soliton increase due to the increase of the nonlinear interactions of exciton-phonon interaction, i.e., the new wave function with two-quanta state and the new Hamiltonian with the added interaction produce considerable changes to the properties of the soliton. In fact, the nonlinear interaction energy in the new model is Gp=8(χ12)2 /(1-s2)w=3.8×10-21J, and it is larger than the linear dispersion energy, J=1.55×10-22J, i.e., the nonlinear interaction is so large that it can really cancel or suppress the linear dispersion effects in the equation of motion of this model. From this point the soliton is stable according to the conditions of formation and stability of the soliton in the soliton theory [27,28]. By comparison, the non-linear interaction energy in the Davydov model is GD=4χ21/(1-s2)w≈1.18×10-21J and it is 3-4 times smaller than Gp. Thus the stability of the Davydov soliton is weak compared to that of the new soliton. Moreover, the binding energy of the quaasi-coherent soliton in Pang’s model is EBP=4 μ J/3=7.8×10- 21J in Eq.(19), which is about 2 times larger than the thermal energy, KBT=4.14×10-21J, at 300K, and about 6 times larger than the Debye energy, K B Θ= ω D MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGcbaabeaakiabfI5arbbaaaaaaaaapeGaeyypa0Zdaiab l+qiO9qacqaHjpWDpaWaaSbaaSqaa8qacaWGebaapaqabaaaaa@3E97@ =1.2×10-21J (here ωD is Debye frequency), and it is approximately equal to ε 0 /4=8.2× 10 21 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaOGaai4laiaaisdaqaaaaaaaaaWdbiabg2da 9iaaiIdacaGGUaGaaGOmaiabgEna0kaaigdacaaIWaWdamaaCaaale qabaWdbiabgkHiTaaak8aadaahaaWcbeqaa8qacaaIYaGaaGymaaaa kiaadQeaaaa@44BE@ , i.e., it has same order of magnitude of the energy of the amide-I vibrational quantum, ε 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaaaa@3882@ ./T4his shows that the quasi-coherent soliton is robust against the quantum fluctuation and thermal perturbation of the systems due to the large energy gap between the soliton state and the delocalized state. In contrast, the binding energy of the Davydov soliton is only EBD= χ 1 4 3 w 2 J MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHhpWydaqhaaWcbaGaaGymaaqaaiaaisdaaaaakeaacaaIZaGaam4D amaaCaaaleqabaGaaGOmaaaakiaadQeaaaaaaa@3CE7@ =0.188×10-21J , which is about 41 times smaller than that of the new soliton, about 23 times smaller than KBT and about six times smaller than K B Θ= ω D MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGcbaabeaakiabfI5arbbaaaaaaaaapeGaeyypa0Zdaiab l+qiO9qacqaHjpWDpaWaaSbaaSqaa8qacaWGebaapaqabaaaaa@3E97@ =1.2×10-21J ,respectively. Therefore, it is easily destroyed by thermal and quantum effects. Hence the Davydov soliton has very small lifetime (about 10-12 ˜ 0-13s), and it is unstable at 300 K [15-18,24- 26]. Therefore, the quasi-coherent soliton can provide a realistic mechanism for the bio-energy transport in protein molecules.

The two-quanta nature of the quasi-coherent soliton plays a more important role in the increase of lifetime relative to that of the added interaction because of the following facts. (1) The change of the nonlinear interaction energyGP=2GD [ 1+2( χ 2 χ 1 )+ ( χ 2 χ 1 ) 2 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaa qabaGcdaWadaqaaiaaigdacqGHRaWkcaaIYaWaaeWaaeaadaWcaaqa aiabeE8aJnaaBaaaleaaieaacaWFYaaabeaaaOqaaiabeE8aJnaaBa aaleaacaWFXaaabeaaaaaakiaawIcacaGLPaaacqGHRaWkdaqadaqa amaalaaabaGaeq4Xdm2aaSbaaSqaaiaa=jdaaeqaaaGcbaGaeq4Xdm 2aaSbaaSqaaiaa=fdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaaaOGaay5waiaaw2faaaaa@4A04@ by μp produced the added interaction in the new model are ΔG =GP (χ2≠0) (χ2=0)=1.08GD < GP(χ2=0)=2GD and Δμ=μP(χ2≠0)-μP(χ2=0)=1.08μD< μP(χ2=0) =2μD,respectively, i.e., the roles of the added interaction on Gp and μp are smaller than that of the two-quanta nature. The two parameters GP and μP are responsible for the lifetime of the soliton. Thus the effect of the former on the lifetimes is smaller than the latter. (2) The contribution of the added interaction to the binding energy of the soliton is about E BP = E BD [ 1+( χ 2 χ 1 ) ] 4 =2.6 E BD MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaadkeacaWGqbaabeaakiabg2da9iaadweadaWgaaWc baGaamOqaiaadseaaeqaaOWaamWaaeaacaaIXaGaey4kaSYaaeWaae aadaWcaaqaaiabeE8aJnaaBaaaleaacaaIYaaabeaaaOqaaiabeE8a JnaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPaaaaiaawUfaca GLDbaadaahaaWcbeqaaiaaisdaaaGccqGH9aqpcaaIYaGaaiOlaiaa iAdacaWGfbWaaSbaaSqaaiaadkeacaWGebaabeaaaaa@4D5B@ which is smaller than that of the two-quanta nature which is E =16EBD. Putting them together in Eq.(35) we see that EBP≈ 41EBD. (3)From the (χ12)- dependence of τ/τ0 in Fig.5, τ/τ0≈100 has already been found directly at χ2=0 which is about 20 times larger than that of the Davydov soliton under the same conditions. This shows clearly that the major effect in the increase of the lifetime is due to the modified wave function. Therefore, it is very reasonable to refer to the new soliton as the quasi-coherent soliton [30-35].

The above calculation helps to resolve the controversies on the lifetime of the Davydov soliton, which is too small in the region of biological temperature. However, by modifying the wave function and the Hamiltonian of the model, a stable soliton at biological temperatures could be produced. This result was obtained considering a new coupled interaction between the acoustic and amide-I vibration modes and a wave function with quasi-coherent two-quanta state. In such a way, the quasi-coherent soliton is a viable mechanism for the bio-energy transport in living systems. Therefore, it can be seen that Pang’s model is completely different from the Davydov’s model. Thus, the equation of motion and properties of the soliton occurring in Pang’s model are also different from that in the Davydov’s model. The distinction of features of the solitons between the two models is shown in Table 3[15]. From the table 3 we know that our new model repulse and refuse the shortcomings of the Davydov model [3], the new soliton in Pang’s model is thermal stable at biological temperature 300K, and has so enough long lifetime, thus it can plays important role in biological processes.
Appendix A
The partial diagonalization of the Hamiltonian implies the diagonalization of that part of the Hamiltonian in Eq. (50) which does not involve the creation and annihilation operators of new phonons Eq.(48). Thus the condition imposed into the functions Cj(x) contained in Eq.(53) to realize such a diagonalization are equivalent, in the continuum approximation, to the following problems of eigenfunctions Cj(x) and eigenvalues Ej determined by 2[ J r 0 2 2 x 2 +iv x + ε 0 2J+V(x) ] C j (x)= E j C j (x)      (A1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaadm aabaGaeyOeI0IaamOsaiaadkhadaqhaaWcbaGaaGimaaqaaiaaikda aaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaaaOqaaiabgk Gi2kaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaamyAaiab l+qiOjaadAhadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhaaaGaey 4kaSIaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaGOmaiaa dQeacqGHRaWkcaWGwbGaaiikaiaadIhacaGGPaaacaGLBbGaayzxaa Gaam4qamaaBaaaleaacaWGQbaabeaakiaacIcacaWG4bGaaiykaiab g2da9iaadweadaWgaaWcbaGaamOAaaqabaGccaWGdbWaaSbaaSqaai aadQgaaeqaaOGaaiikaiaadIhacaGGPaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeikaiaabgeacaqGXaGaaeykaaaa@66AF@ For the above expression of V(x) in Eq.(52) there is only one bound state in Eq.(A1) C s (x)= ( μ p 2 r 0 ) 1/2 sech( μ p x/ r 0 ) exp[ivx/2J r 0 2 ]       (A2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbaabeaaieaakiaa=HcacaWG4bGaa8xkaiabg2da9maa bmaabaWaaSaaaeaacqaH8oqBdaWgaaWcbaGaamiCaaqabaaakeaaca WFYaGaamOCamaaBaaaleaacaWFWaaabeaaaaaakiaawIcacaGLPaaa daahaaWcbeqaaiaa=fdacaWFVaGaa8Nmaaaakiaa=bcacaWFGaGaa8 3Caiaa=vgacaWFJbGaa8hAaiaa=HcacqaH8oqBdaWgaaWcbaGaamiC aaqabaGccaWF4bGaa83laiaa=jhadaWgaaWcbaGaa8hmaaqabaGcca WFPaGaa8hiaiaa=vgacaWF4bGaa8hCaiaa=TfacaWFPbGaeS4dHGMa a8NDaiaa=HhacaWFVaGaa8Nmaiaa=PeacaWFYbWaa0baaSqaaiaa=b daaeaacaWFYaaaaOGaa8xxaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGOaGaaeyqaiaabkdacaqGPaaaaa@6670@ with energy E s =2[ ε 0 2J 2 v 2 4J r 0 2 J μ p 2 ]       (A3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGZbaabeaakiabg2da9iaaikdadaWadaqaaiabew7aLnaa BaaaleaacaaIWaaabeaakiabgkHiTiaaikdacaWGkbGaeyOeI0YaaS aaaeaacqWIpecAdaahaaWcbeqaaiaaikdaaaGccaWG2bWaaWbaaSqa beaacaaIYaaaaaGcbaGaaGinaiaadQeacaWGYbWaa0baaSqaaiaaic daaeaacaaIYaaaaaaakiabgkHiTiaadQeacqaH8oqBdaqhaaWcbaGa amiCaaqaaiaaikdaaaaakiaawUfacaGLDbaacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgeacaqGZaGaaeyk aaaa@5688@ and unbounded(delocalized) states C k (x)= μ p tanh( μ p x/ r 0 )ik r 0 N r 0 [ μ p ik r 0 ] exp[ikx+ivx/2J r 0 2 ]       (A4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGRbaabeaaieaakiaa=HcacaWG4bGaa8xkaiabg2da9maa laaabaGaeqiVd02aaSbaaSqaaiaadchaaeqaaOGaciiDaiaacggaca GGUbGaaiiAaiaa=HcacqaH8oqBdaWgaaWcbaGaamiCaaqabaGccaWG 4bGaa83laiaadkhadaWgaaWcbaGaa8hmaaqabaGccaWFPaGaeyOeI0 IaamyAaiaadUgacaWGYbWaaSbaaSqaaiaa=bdaaeqaaaGcbaWaaOaa aeaacaWGobGaamOCamaaBaaaleaacaWFWaaabeaaaeqaaOGaa83wai abeY7aTnaaBaaaleaacaWGWbaabeaakiabgkHiTiaadMgacaWGRbGa amOCamaaBaaaleaacaWFWaaabeaakiaa=1faaaGaciyzaiaacIhaca GGWbGaa83waiaadMgacaWGRbGaamiEaiabgUcaRiaadMgacqWIpecA caWG2bGaamiEaiaa=9cacaWFYaGaamOsaiaadkhadaqhaaWcbaGaa8 hmaaqaaiaa=jdaaaGccaGGDbGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabIcacaqGbbGaaeinaiaabMcaaaa@72CC@ with energy E k =2[ ε 0 2J 2 v 2 2J r 0 2 +J (k r 0 ) 2 ]       (A5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGRbaabeaakiabg2da9Gqaaiaa=jdadaWadaqaaiabew7a LnaaBaaaleaacaWFWaaabeaakiabgkHiTiaa=jdacaWGkbGaeyOeI0 YaaSaaaeaacqWIpecAdaahaaWcbeqaaiaa=jdaaaGccaWG2bWaaWba aSqabeaacaaIYaaaaaGcbaGaa8NmaiaadQeacaWGYbWaa0baaSqaai aa=bdaaeaacaWFYaaaaaaakiabgUcaRiaadQeacaGGOaGaam4Aaiaa dkhadaWgaaWcbaGaaGimaaqabaGccaGGPaWaaWbaaSqabeaacaWFYa aaaaGccaGLBbGaayzxaaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabIcacaqGbbGaaeynaiaabMcaaaa@57BB@ The energy of the lowest unbounded state is greater than that of the bounded state by the value . The functions Ck(x) are normalized as follows: dx C k (x) C k (x)=δ(k r 0 k r 0 ), dx| C k (x) | 2 =1, dx C s (x) C k (x)=0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGKbGaamiEaiaadoeadaqhaaWcbaGaam4AaaqaaiabgEHiQaaaieaa kiaa=HcacaWG4bGaa8xkaiaadoeadaWgaaWcbaGabm4Aayaafaaabe aaaeaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIipakiaa=Hca caWG4bGaa8xkaiabg2da9iabes7aKjaa=HcacaWGRbGaamOCamaaBa aaleaacaWFWaaabeaakiabgkHiTiqadUgagaqbaiaadkhadaWgaaWc baGaa8hmaaqabaGccaWFPaGaa8hlaiaa=bcadaWdXaqaaiaadsgaca WG4bGaa8hFaiaadoeadaqhaaWcbaGaam4Aaaqaaaaakiaa=HcacaWG 4bGaa8xkaiaa=XhadaahaaWcbeqaaiaa=jdaaaGccqGH9aqpcaWFXa Gaa8hlaiaa=bcaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGH RiI8aOWaa8qmaeaacaWGKbGaamiEaiaadoeadaqhaaWcbaGaam4Caa qaaiabgEHiQaaakiaa=HcacaWG4bGaa8xkaiaadoeadaWgaaWcbaGa am4AaaqabaGccaWFOaGaamiEaiaa=LcacqGH9aqpcaWFWaaaleaacq GHsislcqGHEisPaeaacqGHEisPa0Gaey4kIipaaaa@7844@ Therefore, A+s is an excitation which is localized at the lattice distortion, while A+k creates an unbounded excitation with wave vector k.

In getting Eq. (A1) the variable x was assumed to be continuous and the chain length to tend to infinity L=Nr0→∞. Thus this wave vector k has a continuous value between -∞ and ∞. In subsequent calculation we mainly use a discrete description. The continuous description is transformed into a discrete one according to the rules dx/ r 0 n , dx 2π N r 0 k , δ(k r 0 - k r 0 ) N 2π δ k k , C s (x) C s (n), C k (x) ( N 2π ) 1/2 C k (n) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGKbGaamiEaGqaaiaa=9cacaWGYbWaaSbaaSqaaiaa=bdaaeqaaaqa aiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyOKH46aaa buaeaacaWFSaaaleaacaWGUbaabeqdcqGHris5aOGaa8hiaiaa=bca daWdXaqaaiaadsgacaWG4bGaeyOKH46aaSaaaeaacaWFYaGaeqiWda habaGaamOtaiaadkhadaWgaaWcbaGaa8hmaaqabaaaaOWaaabuaeaa caWFSaaaleaacaWGRbaabeqdcqGHris5aaWcbaGaeyOeI0IaeyOhIu kabaGaeyOhIukaniabgUIiYdGccaWFGaGaa8hiaiabes7aKjaa=Hca caWFRbGaa8NCamaaBaaaleaacaWFWaaabeaakiaa=1caceWFRbGbau aacaWFYbWaaSbaaSqaaiaa=bdaaeqaaOGaa8xkaiabgkziUoaalaaa baGaa8Ntaaqaaiaa=jdacqaHapaCaaGaeqiTdq2aaSbaaSqaaiaa=T gaceWFRbGbauaaaeqaaOGaa8hlaiaa=bcacaWFGaGaa83qamaaBaaa leaacaWFZbaabeaakiaa=HcacaWG4bGaa8xkaiabgkziUkaadoeada WgaaWcbaGaam4CaaqabaGccaWFOaGaamOBaiaa=LcacaWFSaGaam4q amaaBaaaleaacaWGRbaabeaakiaa=HcacaWG4bGaa8xkaiabgkziUo aabmaabaWaaSaaaeaacaWGobaabaGaa8Nmaiabec8aWbaaaiaawIca caGLPaaadaahaaWcbeqaaiaa=fdacaWFVaGaa8Nmaaaakiaadoeada WgaaWcbaGaam4AaaqabaGccaWFOaGaamOBaiaa=Lcaaaa@8C26@ Utilizing Eqs. (50)-(51), (53) and (54), then the partially diagonalized Hamiltonian in the new representation is just Eq.(55).
Appendix B
We now calculate U(k, k’’,t) in Eq.(72) utilizing the coherent state |u> defined by bq|u>=uq|u> with <u| u >=exp{ q [ u q * u q 1 2 | u q | 2 1 2 | u q | 2 ] },|u>=exp q ( u q b q + u q * b q ) ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam yDaGqaaiaa=XhaceWG1bGbauaacqGH+aGpcqGH9aqpciGGLbGaaiiE aiaacchadaGadaqaamaaqafabaWaamWaaeaacaWG1bWaa0baaSqaai aadghaaeaacaWFQaaaaOGabmyDayaafaWaaSbaaSqaaiaadghaaeqa aOGaeyOeI0YaaSaaaeaacaWFXaaabaGaa8NmaaaacaWF8bGaamyDam aaBaaaleaacaWGXbaabeaakiaa=XhadaahaaWcbeqaaiaa=jdaaaGc cqGHsisldaWcaaqaaiaa=fdaaeaacaWFYaaaaiaa=XhacaWG1bWaaS baaSqaaiaadghaaeqaaOGaa8hFamaaCaaaleqabaGaa8NmaaaaaOGa ay5waiaaw2faaaWcbaGaamyCaaqab0GaeyyeIuoaaOGaay5Eaiaaw2 haaiaa=XcacaWF8bGaamyDaiabg6da+iabg2da9iGacwgacaGG4bGa aiiCamaaqafabaGaa8hkaiaadwhadaWgaaWcbaGaamyCaaqabaGcca WGIbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccqGHsislcaWG1bWa a0baaSqaaiaadghaaeaacaWFQaaaaOGaamOyamaaDaaaleaacaWGXb aabaaaaOGaa8xkaaWcbaGaamyCaaqab0GaeyyeIuoakiaa=1faaaa@71DA@ Utilizing the coherent state |u>, the U(k,k″,t) in Eq.(72) can be represented by U(k, k ,t)= 1 Z ph dΩ(u) dΩ( u )( u k * + u k ) ( u k * + u k )<u|exp q ( ω q qv)(β+it) b q + b q }| u > MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaiab g2da9maalaaabaGaa8xmaaqaaiaadQfadaWgaaWcbaGaamiCaiaadI gaaeqaaaaakmaapeaabaGaamizaiabfM6axjaa=HcacaWG1bGaa8xk aaWcbeqab0Gaey4kIipakmaapeaabaGaamizaiabfM6axjaa=Hcace WG1bGbayaacaWFPaGaa8hkaiqadwhagaGbamaaDaaaleaacaWGRbaa baGaa8NkaaaakiabgUcaRiqadwhagaGbamaaDaaaleaacqGHsislca WGRbaabaaaaOGaa8xkaaWcbeqab0Gaey4kIipakiaa=HcacaWG1bWa a0baaSqaaiabgkHiTiqadUgagaGbaaqaaiaa=PcaaaGccqGHRaWkca WG1bWaa0baaSqaaiabgkHiTiqadUgagaGbaaqaaaaakiaa=LcacqGH 8aapcaWG1bGaa8hFaiGacwgacaGG4bGaaiiCamaaqafabaGaa8hkai abeM8a3naaBaaaleaacaWGXbaabeaakiabgkHiTiaadghacaWG2bGa a8xkaiaa=HcacqGHsislcqaHYoGycqWIpecAcqGHRaWkcaWGPbGaam iDaiaa=LcacaWGIbWaa0baaSqaaiaadghaaeaacqGHRaWkaaGccaWG IbWaaSbaaSqaaiaadghaaeqaaOGaa8xFaiaa=XhaceWG1bGbayaacq GH+aGpaSqaaiaadghaaeqaniabggHiLdGccqGHflY1aaa@835B@ < u |exp{ i q ( ω q qv)[ ( b q + b q + 1 n N ( b q + α q + α q * b q )+ 1 n 2 1 N | α q | 2 ]t }|u>    (B1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJabm yDayaagaacbaGaa8hFaiGacwgacaGG4bGaaiiCamaacmaabaGaeyOe I0IaamyAamaaqafabaGaa8hkaiabeM8a3naaBaaaleaacaWGXbaabe aakiabgkHiTiaadghacaWG2bGaa8xkamaadmaabaGaa8hkaiaadkga daqhaaWcbaGaamyCaaqaaiabgUcaRaaakiaadkgadaWgaaWcbaGaam yCaaqabaGccqGHRaWkdaWcaaqaaiaa=fdaaeaacaWFUbWaaOaaaeaa caWGobaaleqaaaaakiaa=HcacaWGIbWaa0baaSqaaiaadghaaeaacq GHRaWkaaGccqaHXoqydaWgaaWcbaGaamyCaaqabaGccqGHRaWkcqaH XoqydaqhaaWcbaGaamyCaaqaaiaa=PcaaaGccaWGIbWaaSbaaSqaai aadghaaeqaaOGaa8xkaiabgUcaRmaalaaabaGaa8xmaaqaaiaa=5ga daahaaWcbeqaaiaa=jdaaaaaaOWaaSaaaeaacaWFXaaabaGaamOtaa aacaWF8bGaeqySde2aaSbaaSqaaiaadghaaeqaaOGaa8hFamaaCaaa leqabaGaa8NmaaaaaOGaay5waiaaw2faaiaadshaaSqaaiaadghaae qaniabggHiLdaakiaawUhacaGL9baacaWF8bGaamyDaiabg6da+iaa bccacaqGGaGaaeiiaiaabccacaqGOaGaaeOqaiaabgdacaqGPaaaaa@76FE@ Where the integration measure is defined as dΩ(u)= k 1 π d x k d y k   ,with  x k +i y k = u k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabfM 6axjaacIcacaWG1bGaaiykaiabg2da9maaxababaGaey4dIunaleaa caWGRbaabeaakmaalaaabaGaaGymaaqaaiabec8aWbaacaWGKbGaam iEamaaBaaaleaacaWGRbaabeaakiaadsgacaWG5bWaaSbaaSqaaiaa dUgaaeqaaOGaaeiiaiaabccacaGGSaGaam4DaiaadMgacaWG0bGaam iAaiaabccacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSIaamyA aiaadMhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaWG1bWaaSbaaS qaaiaadUgaaeqaaaaa@56C7@ Since we can show that exp(τ b k + b k )| u k >=exp{ 1 2 | u k | 2 ( e τ+ τ * 1) }| e τ u k > MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacI hacaGGWbacbaGaa8hkaiabes8a0jaadkgadaqhaaWcbaGaam4Aaaqa aiabgUcaRaaakiaadkgadaWgaaWcbaGaam4AaaqabaGccaWFPaGaa8 hFaiaadwhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpcqGH9aqpcaWG LbGaamiEaiaa=bhadaGadaqaamaalaaabaGaa8xmaaqaaiaa=jdaaa Gaa8hFaiaadwhadaWgaaWcbaGaam4AaaqabaGccaWF8bWaaWbaaSqa beaacaWFYaaaaOGaa8hkaiaadwgadaahaaWcbeqaaiabes8a0jabgU caRiabes8a0naaCaaameqabaGaa8NkaaaaaaGccqGHsislcaWFXaGa a8xkaaGaay5Eaiaaw2haaiaa=XhacaWGLbWaaWbaaSqabeaacqaHep aDaaGccaWG1bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@61FE@ it follows that the first matrix element in Eq.(B1) equals < u k |exp[ q ( ω q qv)(β+it) b q + b q ]| u k >= MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam yDamaaBaaaleaacaWGRbaabeaakiaacYhaciGGLbGaaiiEaiaaccha caGGBbWaaabuaeaacaGGOaGaeqyYdC3aaSbaaSqaaiaadghaaeqaaO GaeyOeI0IaamyCaiaadAhacaGGPaGaaiikaiabgkHiTiabek7aIjab l+qiOjabgUcaRiaadMgacaWG0bGaaiykaiaadkgadaqhaaWcbaGaam yCaaqaaiabgUcaRaaakiaadkgadaWgaaWcbaGaamyCaaqabaGccaGG DbGaaiiFaiqadwhagaGbamaaBaaaleaacaWGRbaabeaaaeaacaWGXb aabeqdcqGHris5aOGaeyOpa4Jaeyypa0daaa@5B0F@ exp{ k ( 1 2 | u k | 2 + 1 2 | u k | 2 u k * u k exp[( ω q qv)(β+it)] ) } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacI hacaGGWbWaaiWaaeaacqGHsisldaaeqbqaamaabmaabaWaaSaaaeaa caaIXaaabaGaaGOmaaaacaGG8bGaamyDamaaBaaaleaacaWGRbaabe aakiaacYhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaWcaaqaaiaa igdaaeaacaaIYaaaaiaacYhaceWG1bGbayaadaWgaaWcbaGaam4Aaa qabaGccaGG8bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyDamaa DaaaleaacaWGRbaabaGaaiOkaaaakiqadwhagaGbamaaBaaaleaaca WGRbaabeaakiGacwgacaGG4bGaaiiCaiaacUfacaGGOaGaeqyYdC3a aSbaaSqaaiaadghaaeqaaOGaeyOeI0IaamyCaiaadAhacaGGPaGaai ikaiabgkHiTiabek7aIjabl+qiOjabgUcaRiaadMgacaWG0bGaaiyk aiaac2faaiaawIcacaGLPaaaaSqaaiaadUgaaeqaniabggHiLdaaki aawUhacaGL9baaaaa@6844@ The second matrix element in Eq. (B1) can be represented as a path integral that can be evaluated exactly. Utilizing the general relationship between the matrix element and the path integral: < u k |exp[iω( b k + b k + τ * b k + b k + τ+ τ * τ]}| u k > =exp[ 1 2 (| u k | 2 +| u k | 2 iω|τ | 2 t] y*(t)= u * y(0)= u q D(y*,y)exp[iT(y*,y)]     (B2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqGH8a apceWG1bGbayaadaWgaaWcbaGaam4AaaqabaacbaGccaWF8bGaciyz aiaacIhacaGGWbGaa83waiabgkHiTiaadMgacqaHjpWDcaWFOaGaam OyamaaDaaaleaacaWGRbaabaGaey4kaScaaOGaamOyamaaBaaaleaa caWGRbaabeaakiabgUcaRiabes8a0naaCaaaleqabaGaa8Nkaaaaki aadkgadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaWGIbWaa0baaSqa aiaadUgaaeaacqGHRaWkaaGccqaHepaDcqGHRaWkcqaHepaDdaahaa Wcbeqaaiaa=PcaaaGccqaHepaDcaWFDbGaa8xFaiaa=XhacaWG1bWa aSbaaSqaaiaadUgaaeqaaOGaeyOpa4dabaGaeyypa0JaciyzaiaacI hacaGGWbGaa83waiabgkHiTmaalaaabaGaa8xmaaqaaiaa=jdaaaGa a8hkaiaa=XhaceWG1bGbayaadaWgaaWcbaGaam4AaaqabaGccaWF8b WaaWbaaSqabeaacaWFYaaaaOGaey4kaSIaa8hFaiaadwhadaWgaaWc baGaam4AaaqabaGccaWF8bWaaWbaaSqabeaacaWFYaaaaOGaeyOeI0 IaamyAaiabeM8a3jaa=XhacqaHepaDcaWF8bWaaWbaaSqabeaacaWF YaaaaOGaamiDaiaa=1fadaWdXaqaaiaadseacaWFOaGaamyEaiaa=P cacaWFSaGaamyEaiaa=LcaciGGLbGaaiiEaiaacchacaWFBbGaamyA aiaadsfacaWFOaGaamyEaiaa=PcacaWFSaGaamyEaiaa=LcacaWFDb aaleaacaWG5bGaa8Nkaiaa=HcacaWG0bGaa8xkaiabg2da9iqadwha gaGbaiaa=PcaaeaadaWgaaadbaGaamyEaiaa=HcacaWFWaGaa8xkai abg2da9iaadwhadaWgaaqaaiaadghaaeqaaaqabaaaniabgUIiYdGc caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkeacaqGYaGaaeykaa aaaa@9F94@ where T(y*,y)= 0 t d t { iy*( t ) dy d t ω[y*(t)y( t )+τ*y( t )+y*( t )τ] }i u k *y(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaacI cacaWG5bGaaiOkaiaacYcacaWG5bGaaiykaiabg2da9maapedabaGa amizaiqadshagaqbamaacmaabaGaamyAaiaadMhacaGGQaGaaiikai qadshagaqbaiaacMcadaWcaaqaaiaadsgacaWG5baabaGaamizaiqa dshagaqbaaaacqGHsislcqaHjpWDcaGGBbGaamyEaiaacQcacaGGOa GaamiDaiaacMcacaWG5bGaaiikaiqadshagaqbaiaacMcacqGHRaWk cqaHepaDcaGGQaGaamyEaiaacIcaceWG0bGbauaacaGGPaGaey4kaS IaamyEaiaacQcacaGGOaGabmiDayaafaGaaiykaiabes8a0jaac2fa aiaawUhacaGL9baacqGHsislcaWGPbGabmyDayaagaWaa0baaSqaai qadUgagaqbaaqaaaaakiaacQcacaWG5bGaaiikaiaadshacaGGPaaa leaacaaIWaaabaGaamiDaaqdcqGHRiI8aaaa@6EA8@ We can evaluate the path integral by stardand techniques. The result for Eq. (B2) is { 1 2 (| u k | 2 +| u k | 2 + u k * u k e iωt (1 e iωt )( u t *τ+τ* u k +|τ | 2 )}   (B3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa83Eai abgkHiTmaalaaabaGaa8xmaaqaaiaa=jdaaaGaa8hkaiaa=XhaceWG 1bGbayaadaWgaaWcbaGaam4AaaqabaGccaWF8bWaaWbaaSqabeaaca WFYaaaaOGaey4kaSIaa8hFaiaadwhadaWgaaWcbaGaam4AaaqabaGc caWF8bWaaWbaaSqabeaacaWFYaaaaOGaey4kaSIabmyDayaagaWaaS baaSqaaiaadUgaaeqaaOGaa8NkaiaadwhadaWgaaWcbaGaam4Aaaqa baGccaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdCNaamiDaa aakiabgkHiTiaa=HcacaWFXaGaeyOeI0IaamyzamaaCaaaleqabaGa eyOeI0IaamyAaiabeM8a3jaadshaaaGccaWFPaGaa8hkaiqadwhaga GbamaaBaaaleaacaWG0baabeaakiaa=PcacqaHepaDcqGHRaWkcqaH epaDcaWFQaGaamyDamaaBaaaleaacaWGRbaabeaakiabgUcaRiaa=X hacqaHepaDcaWF8bWaaWbaaSqabeaacaWFYaaaaOGaa8xkaiaa=1ha caqGGaGaaeiiaiaabccacaqGOaGaaeOqaiaabodacaqGPaaaaa@70E8@ Substituting above the matrix elements obtained into Eq.(B1) we get U(k, k ,t)= e R n (t) Z ph d Ω(u) dΩ( u )( u k *+ u k )( u k *+ u k ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaGqaai aa=HcacaWGRbGaa8hlaiqadUgagaGbaiaa=XcacaWG0bGaa8xkaiab g2da9maalaaabaGaamyzamaaCaaaleqabaGaamOuamaaBaaameaaca WGUbaabeaaliaa=HcacaWG0bGaa8xkaaaaaOqaaiaadQfadaWgaaWc baGaamiCaiaadIgaaeqaaaaakmaapeaabaGaamizaaWcbeqab0Gaey 4kIipakiabfM6axjaa=HcacaWG1bGaa8xkamaapeaabaGaamizaiab fM6axjaa=HcaceWG1bGbayaacaWFPaGaa8hkaiqadwhagaGbamaaBa aaleaacaWGRbaabeaakiaa=PcacqGHRaWkceWG1bGbayaadaWgaaWc baGaeyOeI0Iaam4AaaqabaGccaWFPaGaa8hkaiaadwhadaWgaaWcba GaeyOeI0Iabm4Aayaagaaabeaakiaa=PcacqGHRaWkcaWG1bWaaSba aSqaaiqadUgagaGbaaqabaGccaWFPaaaleqabeqdcqGHRiI8aaaa@63CA@ exp{ q ( | u q | 2 +| u q | 2 u q * u q exp[( ω q qv)(β+it)] u q * u q exp[i( ω q qv)t]+ 1 n 1 N ( u q α q + u q * α q * )(1exp[i( ω q qv )t]) ) }    (B4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGLb GaaiiEaiaacchadaGabaqaaiabgkHiTmaaqafabaacbaGaa8hkaaWc baGaamyCaaqab0GaeyyeIuoaaOGaay5EaaGaa8hFaiaadwhadaWgaa WcbaGaamyCaaqabaGccaWF8bWaaWbaaSqabeaacaWFYaaaaOGaey4k aSIaa8hFaiqadwhagaGbamaaBaaaleaacaWGXbaabeaakiaa=Xhada ahaaWcbeqaaiaa=jdaaaGccqGHsislcaWG1bWaa0baaSqaaiaadgha aeaacaWFQaaaaOGabmyDayaagaWaaSbaaSqaaiaadghaaeqaaOGaci yzaiaacIhacaGGWbGaa83waiaa=HcacqaHjpWDdaWgaaWcbaGaamyC aaqabaGccqGHsislcaWGXbGaamODaiaa=LcacaWFOaGaeyOeI0Iaeq OSdiMaeS4dHGMaey4kaSIaamyAaiaadshacaWFPaGaa8xxaiabgkHi TiqadwhagaGbamaaBaaaleaacaWGXbaabeaakiaa=PcacaWG1bWaa0 baaSqaaiaadghaaeaaaaGccqGHflY1aeaaciGGLbGaaiiEaiaaccha caWFBbGaeyOeI0IaamyAaiaa=HcacqaHjpWDdaWgaaWcbaGaamyCaa qabaGccqGHsislcaWGXbGaamODaiaa=LcacaWG0bGaa8xxaiabgUca RmaalaaabaGaa8xmaaqaaiaa=5gaaaWaaiGaaeaadaWcaaqaaiaa=f daaeaadaGcaaqaaiaad6eaaSqabaaaaOGaa8hkaiqadwhagaGbamaa BaaaleaacaWGXbaabeaakiabeg7aHnaaBaaaleaacaWGXbaabeaaki abgUcaRiaadwhadaqhaaWcbaGaamyCaaqaaiaa=PcaaaGccqaHXoqy daqhaaWcbaGaamyCaaqaaiaa=PcaaaGccaWFPaGaa8hkaiaa=fdacq GHsislciGGLbGaaiiEaiaacchacaWFBbGaamyAaiaa=HcacqaHjpWD daWgaaWcbaGaamyCaaqabaGccqGHsislcaWGXbGaamODamaabiaaba Gaa8xkaiaadshacaWFDbGaa8xkaaGaayzkaaaacaGL9baacaqGGaGa aeiiaiaabccacaqGGaGaaeikaiaabkeacaqG0aGaaeykaaaaaa@A3AE@ where R n ( t )= 1 n 2 N k | α k | 2 (1exp[i( ω k kv)t])      (B5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaamOBaaWdaeqaaOWaaeWaaeaapeGa amiDaaWdaiaawIcacaGLPaaapeGaeyypa0ZdamaalaaabaGaeyOeI0 IaaGymaaqaaiaad6gadaahaaWcbeqaaiaaikdaaaGccaWGobaaamaa qafabaGaaiiFaiabeg7aHnaaBaaaleaacaWGRbaabeaakiaacYhada ahaaWcbeqaaiaaikdaaaGccaGGOaGaaGymaiabgkHiTiGacwgacaGG 4bGaaiiCaiaacUfacqGHsislcaWGPbGaaiikaiabeM8a3naaBaaale aacaWGRbaabeaakiabgkHiTiaadUgacaWG2bGaaiykaiaadshacaGG DbGaaiykaaWcbaGaam4Aaaqab0GaeyyeIuoakiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeikaiaabkeacaqG1aGaaeykaaaa@60E6@ The and u” integrations can easily be finished. For instance, the contribution from the term with the u k * u k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaga WaaSbaaSqaaiaadUgaaeqaaGqaaOGaa8NkaiaadwhadaWgaaWcbaGa bm4Aayaagaaabeaaaaa@3AF9@ factor, which we can denote by U a (k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGHbaabeaakiaacIcacaWGRbGaaiilaiqadUgagaGbaiaa cYcacaWG0bGaaiykaaaa@3D8A@ since it is associated with the absorption of a phonon, is U a (k, k ,t)= exp[i( ω k kv)t+ R n (t)+ ξ n (t)] exp[β( ω k vk)]1 { δ k k 1 n 2 1 N α k * α k (exp[i( ω k k v)t]1)(exp[i( ω k kv)t]1) exp[β( ω k vk)]1 }      (B6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGvb WaaSbaaSqaaiaadggaaeqaaGqaaOGaa8hkaiaadUgacaWFSaGabm4A ayaagaGaa8hlaiaadshacaWFPaGaeyypa0ZaaSaaaeaaciGGLbGaai iEaiaacchacaWFBbGaamyAaiaa=HcacqaHjpWDdaWgaaWcbaGaam4A aaqabaGccqGHsislcaWGRbGaamODaiaa=LcacaWG0bGaey4kaSIaam OuamaaBaaaleaacaWGUbaabeaakiaa=HcacaWG0bGaa8xkaiabgUca Riabe67a4naaBaaaleaacaWGUbaabeaakiaa=HcacaWG0bGaa8xkai aa=1faaeaaciGGLbGaaiiEaiaacchacaWFBbGaeqOSdiMaeS4dHGMa a8hkaiabeM8a3naaBaaaleaacaWGRbaabeaakiabgkHiTiaadAhaca WGRbGaa8xkaiaa=1facqGHsislcaWFXaaaaiabgkci3cqaamaacmaa baGaeqiTdq2aaSbaaSqaaiaadUgaceWGRbGbayaaaeqaaOGaeyOeI0 YaaSaaaeaadaWcaaqaaiaa=fdaaeaacaWFUbWaaWbaaSqabeaacaWF YaaaaaaakmaalaaabaGaa8xmaaqaaiaad6eaaaGaeqySde2aa0baaS qaaiaadUgaaeaacaWFQaaaaOGaeqySde2aaSbaaSqaaiqadUgagaGb aaqabaGccaWFOaGaciyzaiaacIhacaGGWbGaa83waiaadMgacaWFOa GaeqyYdC3aaSbaaSqaaiqadUgagaGbaaqabaGccqGHsislceWGRbGb ayaacaWG2bGaa8xkaiaadshacaWFDbGaeyOeI0Iaa8xmaiaa=Lcaca WFOaGaciyzaiaacIhacaGGWbGaai4waiabgkHiTiaadMgacaWFOaGa eqyYdC3aaSbaaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4AaiaadAhaca WFPaGaamiDaiaa=1facqGHsislcaWFXaGaa8xkaaqaaiGacwgacaGG 4bGaaiiCaiaa=TfacqaHYoGycqWIpecAcaWFOaGaeqyYdC3aaSbaaS qaaiaadUgaaeqaaOGaeyOeI0IaamODaiaadUgacaWFPaGaa8xxaiab gkHiTiaa=fdaaaaacaGL7bGaayzFaaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeikaiaabkeacaqG2aGaaeykaaaaaa@B375@ where ξ n (t)= 4 n 2 N k | α k | 2 sin 2 ( 1 2 ( ω k vk)t) exp[β( ω k vk)]1      (B7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaad6gaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaa aeaacqGHsislieaacaWF0aaabaGaa8NBamaaCaaaleqabaGaa8Nmaa aakiaa=5eaaaWaaabuaeaadaWcaaqaaiaa=XhacqaHXoqydaWgaaWc baGaam4AaaqabaGccaWF8bWaaWbaaSqabeaacaWFYaaaaOGaci4Cai aacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaaiikamaalaaabaGa aGymaaqaaiaaikdaaaGaaiikaiabeM8a3naaBaaaleaacaWGRbaabe aakiabgkHiTiaadAhacaWGRbGaaiykaiaadshacaGGPaaabaGaciyz aiaacIhacaGGWbGaai4waiabek7aIjabl+qiOjaacIcacqaHjpWDda WgaaWcbaGaam4AaaqabaGccqGHsislcaWG2bGaam4AaiaacMcacaGG DbGaeyOeI0IaaGymaaaaaSqaaiaadUgaaeqaniabggHiLdGccaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGcbGaae4naiaabMca aaa@6E70@ We note that the breaking of the translational symmetry by the deformation leads to off-diagonal terms corresponding to violation of wave vector conservation. However, we can prove that these terms are proportional to 1 N α k * α k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamOtaaaacqaHXoqydaqhaaWcbaGaam4AaaqaaiaacQca aaGccqaHXoqydaWgaaWcbaGabm4Aayaagaaabeaaaaa@3DCF@ which can be neglected when either |k|or |k″| is large as compared to 4μp/πr0 as can be seen in the definition of αk in Eq.(59). Furthermore, when -π≤kr0≤π and μp< π2 the off-diagonal terms are negligible except for a small region at the center of the Brillouin zone. Since the small wave vector terms do not significantly contribute to Γn due to the k-dependence of (q,k), and thus the off-diagonal terms can be neglected in U a (k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGHbaabeaakiaacIcacaWGRbGaaiilaiqadUgagaGbaiaa cYcacaWG0bGaaiykaaaa@3D8A@ in the calculation of Γn. The energy of the soliton state is less than that of the unlocalized exciton in the uniform lattice. Therefore, the parts of corresponding to the absorption of a phonon make the major contributions to the sum in Eq.(72) at the temperature and parameter values of interest, and their off-diagonal terms may also be neglected just as above. Using the result of the U a (k, k ,t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGHbaabeaakiaacIcacaWGRbGaaiilaiqadUgagaGbaiaa cYcacaWG0bGaaiykaaaa@3D8A@ obtained from the above formulae of Eq. (72) the decay rate Eq.(74) can be obtained.
Appendix C
If the soliton velocity approaches zero we can get an analytical expression for R2(t) and ξ 2 (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFC@ at n=2 defined in Eq. (75) or Eqs. (B5) and (B7) through inserting Eq. (59) into Eqs.(B5) and (B7) and applying the relation of 1 N q r 0 2π dq,i.e. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamOtaaaadaaeqbqaaiabgkziUcWcbaGaamyCaaqab0Ga eyyeIuoakmaalaaabaGaamOCamaaBaaaleaacaaIWaaabeaaaOqaai aaikdacqaHapaCaaWaa8qmaeaacaWGKbGaamyCaiaacYcacaWGPbGa aiOlaiaadwgacaGGUaaaleaacqGHsislcqGHEisPaeaacqGHEisPa0 Gaey4kIipaaaa@4CD6@ Lim v0 R 2 (t)= R 0 y s h 2 y { [1cos( ω α ty)]+isin( ω α ty) }dy (here   y= πq r 0 2 μ p )      (C1) = R 0 [i x ψ (1+i x )+ψ(1+i x )ψ(1)] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaadaWfqa qaaiaadYeacaWGPbGaamyBaaWcbaGaamODaiabgkziUIqaaiaa=bda aeqaaOGaamOuamaaBaaaleaacaaIYaaabeaakiaa=HcacaWG0bGaa8 xkaiabg2da9iabgkHiTiaadkfadaWgaaWcbaGaa8hmaaqabaGcdaWd XaqaamaalaaabaGaamyEaaqaaiaadohacaWGObWaaWbaaSqabeaaca WFYaaaaOGaamyEaaaadaGadaqaaiaa=TfacaWFXaGaeyOeI0Iaci4y aiaac+gacaGGZbGaa8hkaiabeM8a3naaBaaaleaacqaHXoqyaeqaaO GaamiDaiaadMhacaWFPaGaa8xxaiabgUcaRiaadMgaciGGZbGaaiyA aiaac6gacaWFOaGaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccaWG0b GaamyEaiaa=LcaaiaawUhacaGL9baacaWGKbGaamyEaiaa=bcacaWF GaGaa8hiaiaa=bcacaWFOaGaamiAaiaadwgacaWGYbGaamyzaiaabc cacaqGGaGaaeiiaiaadMhacqGH9aqpdaWcaaqaaiabec8aWjaadgha caWGYbWaaSbaaSqaaiaa=bdaaeqaaaGcbaGaa8NmaiabeY7aTnaaBa aaleaacaWGWbaabeaaaaGccaWFPaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaaboeacaqGXaGaaeykaaWcbaGaeyOeI0 IaeyOhIukabaGaeyOhIukaniabgUIiYdaakeaacqGH9aqpcqGHsisl caWGsbWaaSbaaSqaaiaa=bdaaeqaaOGaa83waiaadMgaceWG4bGbau aacuaHipqEgaqbaiaa=HcacaWFXaGaey4kaSIaamyAaiqadIhagaqb aiaa=LcacqGHRaWkcqaHipqEcaWFOaGaa8xmaiabgUcaRiaadMgace WG4bGbauaacaWFPaGaeyOeI0IaeqiYdKNaa8hkaiaa=fdacaWFPaGa a8xxaaaaaa@A0B8@ where R 0 = 4 ( χ 1 + χ 2 ) 2 πw ( M w ) 1/2 = 2J μ p r 0 π v 0 , ω α = 2 μ p π ( w M ) 1/2      (C2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaaieaacaWFWaaabeaakiabg2da9maalaaabaGaa8hnaiaa=Hca cqaHhpWydaWgaaWcbaGaa8xmaaqabaGccqGHRaWkcqaHhpWydaWgaa WcbaGaa8NmaaqabaGccaWFPaWaaWbaaSqabeaacaWFYaaaaaGcbaGa eqiWdaNaeS4dHGMaam4DaaaadaqadaqaamaalaaabaGaamytaaqaai aadEhaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWFXaGaa83laiaa =jdaaaGccqGH9aqpdaWcaaqaaiaa=jdacaWGkbGaeqiVd02aaSbaaS qaaiaadchaaeqaaOGaamOCamaaBaaaleaacaWFWaaabeaaaOqaaiab ec8aWjabl+qiOjaadAhadaWgaaWcbaGaa8hmaaqabaaaaOGaa8hlai aa=bcacaWFGaGaa8hiaiaa=bcacqaHjpWDdaWgaaWcbaGaeqySdega beaakiabg2da9maalaaabaGaa8NmaiabeY7aTnaaBaaaleaacaWGWb aabeaaaOqaaiabec8aWbaadaqadaqaamaalaaabaGaam4Daaqaaiaa d2eaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWFXaGaa83laiaa=j daaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGdbGa aeOmaiaabMcaaaa@7087@ ψ is the digamma function, ψ ′is its derivative and x′= t=KBT0t/ℏ

ξ 2 (t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFC@ can be easily elvaluated when v≈0 and R0< 1 at sufficiently high temperature T>T0(T0= ℏ ωα/KB). In this case it is ξ 2 (t)= R 0 ω α [ T T 0 ] 0 d ω k sin 2 [ 1 2 ω k t] s h 2 ( ω k / ω α ) = R 0 T T 0 [1π ω α tcoth(π ω α t)]    (C3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaa aeaacqGHsislcaWGsbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaeqyYdC 3aaSbaaSqaaiabeg7aHbqabaaaaOWaamWaaeaadaWcaaqaaiaadsfa aeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaaaOGaay5waiaaw2faam aapedabaGaamizaiabeM8a3naaBaaaleaacaWGRbaabeaaaeaacaaI WaaabaGaeyOhIukaniabgUIiYdGcdaWcaaqaaiGacohacaGGPbGaai OBamaaCaaaleqabaGaaGOmaaaakiaacUfadaWcaaqaaiaaigdaaeaa caaIYaaaaiabeM8a3naaBaaaleaacaWGRbaabeaakiaadshacaGGDb aabaGaam4CaiaadIgadaahaaWcbeqaaiaaikdaaaGccaGGOaGaeqyY dC3aaSbaaSqaaiaadUgaaeqaaOGaai4laiabeM8a3naaBaaaleaacq aHXoqyaeqaaOGaaiykaaaacqGH9aqpdaWcaaqaaiaadkfadaWgaaWc baGaaGimaaqabaGccaWGubaabaGaamivamaaBaaaleaaieaacaWFWa aabeaaaaGccaGGBbGaaGymaiabgkHiTiabec8aWjabeM8a3naaBaaa leaacqaHXoqyaeqaaOGaamiDaiGacogacaGGVbGaaiiDaiaacIgaca GGOaGaeqiWdaNaeqyYdC3aaSbaaSqaaiabeg7aHbqabaGccaWG0bGa aiykaiaac2facaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaaboeaca qGZaGaaeykaaaa@85C7@ where we use the relation . exp(β ω k )1+β ω k MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacI hacaGGWbGaaiikaiabek7aIjabl+qiOjabeM8a3naaBaaaleaacaWG RbaabeaakiaacMcacqGHijYUcaaIXaGaey4kaSIaeqOSdiMaeS4dHG MaeqyYdC3aaSbaaSqaaiaadUgaaeqaaaaa@48E7@ As t →∞ (because we are interested in the long-time steady behaviour) the leading terms in the above asymptotic formulae of R2(t) and ξ2 (t) can be represented by R 2 (t)= R 0 [ln( 1 2 ω α t)+1.578+ 1 2 iπ]      (C4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIYaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iabgkHi TiaadkfadaWgaaWcbaGaaGimaaqabaGccaGGBbGaciiBaiaac6gaca GGOaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHjpWDdaWgaaWcbaGa eqySdegabeaakiaadshacaGGPaGaey4kaSIaaGymaiaac6cacaaI1a GaaG4naiaaiIdacqGHRaWkdaWcaaqaaGqaaiaa=fdaaeaacaWFYaaa aiaadMgacqaHapaCcaGGDbGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeikaiaaboeacaqG0aGaaeykaaaa@592B@ ξ 2 (t)π R 0 k B Tt/      (C5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaGqaaOGaa8hkaiaadshacaWFPaGaeyisISRa eyOeI0IaeqiWdaNaamOuamaaBaaaleaacaWFWaaabeaakiaadUgada WgaaWcbaGaamOqaaqabaGccaWGubGaamiDaiaa=9cacqWIpecAcaWF GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae4qaiaabw dacaqGPaaaaa@4D55@ (where we approximated coth 1 2 ω α t~1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaie aacaWFXaaabaGaa8NmaaaacqaHjpWDdaWgaaWcbaGaeqySdegabeaa kiaadshacaWF+bGaa8xmaaaa@3DBF@ ) , i.e., Lim t ξ 2 (t)=ηt , η=π R 0 /β=π R 0 k B T/      (C6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaca WGmbGaamyAaiaad2gaaSqaaiaadshacqGHsgIRcqGHEisPaeqaaOGa eqOVdG3aaSbaaSqaaiaaikdaaeqaaGqaaOGaa8hkaiaadshacaWFPa Gaeyypa0JaeyOeI0Iaeq4TdGMaamiDaiaa=bcacaWFSaGaa8hiaiaa =bcacqaH3oaAcqGH9aqpcqaHapaCcaWGsbWaaSbaaSqaaiaa=bdaae qaaOGaa83laiabek7aIjabl+qiOjabg2da9iabec8aWjaadkfadaWg aaWcbaGaa8hmaaqabaGccaWGRbWaaSbaaSqaaiaadkeaaeqaaOGaam ivaiaa=9cacqWIpecAcaWFGaGaa8hiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeikaiaaboeacaqG2aGaaeykaaaa@6380@ Except at low temperature, the x (= ω α t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafa Gaaiikaiabg2da9iabeM8a3naaBaaaleaacqaHXoqyaeqaaOGaamiD aiaacMcaaaa@3DF8@ dependent term in the real part of R2(t) is small with respect to ξ 2 (T) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaaikdaaeqaaOGaaiikaiaadsfacaGGPaaaaa@3ADC@ for parameter values of interest and can be neglected. Furthermore, since R0< 1 ( but it is not very small, about R0≈0.529) and T0< T (but it is not too small, about T0≈294K) and R0 T/T0< 1 for the protein molecules, then one can evaluate the integral in Eq.(72) by using the following approximation and utilizing the above results of Eqs.(C4-C6). 1 π Re 0 dt exp{ i[2J (k' r 0 ) 2 + 4 3 J μ p 2 ω k ]t/+R(t)+ξ(t) } 1 π (2.43 ω α ) R 0 Γ(1 R 0 ) [ η 2 + (δ(k, k )/) 2 ] (1 R 0 )/2 { cos( π R 0 2 ) cos[ (1 R 0 ) tan 1 ( δ(k, k ) η ) ]sin( π R 0 2 )sin [ (1 R 0 ) tan 1 ( δ(k, k ) η ) ] }                 (C7) = 1 π (2.43 ω α ) R 0 Γ(1 R 0 ) [ η 2 + (δ(k, k )/) 2 ] (1 R 0 )/2 cos( Φ 1 + Φ 2 ) 1 π (2.43 ω α ) R 0 Γ(1 R 0 ) [ η 2 + (δ(k, k )/) 2 ] (1 R 0 )/2 [ 1 1 2 [ π R 0 2 +(1 R 0 )( δ(k, k ) η ) ] 2 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x 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Ojaa=LcadaahaaWcbeqaaiaa=jdaaaGccaWFDbWaaWbaaSqabeaacq GHsislcaWFOaGaa8xmaiabgkHiTiaadkfadaWgaaadbaGaa8hmaaqa baWccaWFPaGaa83laiaa=jdaaaGcdaWadaqaaiaa=fdacqGHsislda Wcaaqaaiaa=fdaaeaacaWFYaaaamaadmaabaWaaSaaaeaacqaHapaC caWGsbWaaSbaaSqaaiaa=bdaaeqaaaGcbaGaa8NmaaaacqGHRaWkca WFOaGaa8xmaiabgkHiTiaadkfadaWgaaWcbaGaa8hmaaqabaGccaWF PaWaaeWaaeaadaWcaaqaaiabes7aKjaa=HcacaWGRbGaa8hlaiqadU gagaqbaiaa=LcaaeaacqaH3oaAcqWIpecAaaaacaGLOaGaayzkaaaa caGLBbGaayzxaaWaaWbaaSqabeaacaWFYaaaaaGccaGLBbGaayzxaa aaaaa@6659@ where δ(k, k )=2J ( k r 0 ) 2 + 4 3 μ p 2 J ω k , Φ 1 = R 0 π 2 , Φ 2 =[(1 R 0 ) tan 1 ( δ(k, k ) η )]     (C8) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaai ikaiaadUgacaGGSaGabm4AayaafaGaaiykaiabg2da9iaaikdacaWG 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