Research Article
Open Access

Effects of Maxwellian ions on Dust-acoustic
Solitary Waves in Adiabatic Degenerate Plasmas
and It’s Instability

Nushrat Khan, Sharmin Yiasmin Swarna and Md. Masum Haider

^{*}
Department of Physics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangla

***Corresponding author:**Md. Masum Haider, Department of Physics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh; Email:

Received: October 11, 2017; Accepted: October 25, 2017; Published: November 23, 2017

**Citation:**Haider MD, Khan N, et al. (2017) Effects of Maxwellian ions on Dust-acoustic Solitary Waves in Adiabatic Degenerate Plasmas and It’s Instability. Int J Hematol Blo Dis 1(1): 1-8.

Abstract Top

A rigorous theoretical investigation has been made on
dust acoustic solitary waves in unmagnetized and magnetized
degenerate plasmas with adiabatic pressure. Korteweg-de
Vries (K-dV) equation have been derived for unmagnetized
case as well as its solution. For the case of magnetized plasmas,
Zakharov-Kuznetsov (ZK) equation, and its solution have been
derived and studied its instability criterion and growth rate. It
has been found that, the parametric regimes effects not only
modify the basic properties of dust acoustic solitary waves
and its instability criterion but also introduce some important
new features.

**Keywords:**Maxwellian ions; Dust-acoustic Solitary Waves; Degenerate Plasmas; K-dV equation; ZK euation; Instability;Introduction

The propagation of dust acoustic waves (DASW’s) waves plays
an important role in understanding the different behaviour of
dust components or different waves phenomena in dusty plasmas
which are omnipresent in laboratory, space and astrophysical
plasma environments, such as cometary tails, planetary ring,
interstellar medium etc. The dust grain changes the nature of the
system including the creation of the new modes. In the recent
years, many important and precious, theoretical and experimental
investigation has been made on solitary waves. DASW’s are one of
the most significant non-linear affair in plasma method. Different
properties such as amplitude, width etc. of the solitary waves
can be modified by using different perturbation method. Bliokh
and Yaroshenko studied the elecrostatic waves in dusty plasmas
and applied their results in interpreting spoke-like structures
in Saturn’s rings (revealed by the voyager space mission)[1,
2]. Rao et.al, were the first to predict theoretically existence of
extremely lower phase velocity DASW’s in unmagnetized dusty
plasmas whose constituent are inertial charged dust grains and
Boltzmann distributed ions and electrons [3]. Roychoudhury
and Mukherjee considered a two-component unmagnetized
dusty plasma consisting of a negatively charged adiabatic dust
fluid and an inertia-less isothermal ion fluid and investigated
the effects of dust fluid temperature on large amplitude solitary
waves by the pseudo-potential approach [4]. Sayed and Mamun
assumed a dusty plasma containing the adiabatic dust fluid and
non-adiabatic (isothermal) inertia-less electron and ion fluid and
studied the effect of the dust fluid temperature on the DASW’s by
the reductive perturbation method [5, 6]. Mendoza-Briceño et al,
assumed a two-component dusty plasma containing the adiabatic
dust fluid and non-adiabatic ions following the nonthermal
distribution of Cairns et al and studied the effect of the dust fluid
temperature on the DA solitary waves by the pseudo-potential
approach [7-9].

The equation of state for degenerate electrons in such intersteller compact objects are mathematically explained by Chandrasekhar for two limits, namely non-relativistic and ultrarelativistic limits [10,11]. The degenerate electron equation of state of Chandrasekhar is ${p}_{e}$ is proportional to ${N}_{e}^{5/3}$ for the non-relativistic limit and ${p}_{e}$ is proportional to ${N}_{e}^{4/3}$ the ultrarelativistic limit where ${p}_{e}$ is the degenerate electron pressure and ${N}_{e}$ is the degenerate electron number density. Mamun and Shukla considered an unmagnetized degenerate plasma without and with negatively charged stationary dust [12, 13]. Mamun et al, studied the ion acoustic Solitary Waves (SW’s) in the presence of an external magnetic field for ultra-relativistic degenerate electron-ion plasmas using the reductive perturbation technique [14]. Using the same technique Haider et al, also studied the obliquely propagating solitary structure with the presence of external magnetic field [15,16]. But the presence of heavy ions, which can be arbitrary charged, were not considered in the work of Mamun et al and Haider et al [14,15].

In the present work we have studied the nonlinear propagation of DA SW’s in unmagnetized and magnetized degenerate plasmas with adiabatic pressure. We have studied both ultra-relativistic and non-relativistic case simultaneously using the generalized equation proposed by Haider [17]. To do this we have derived Korteweg-de Vries (K-dV) and Zakharov-Kuznetsov (ZK) equation by reductive perturbation method and find out the solution of it [6]. We have studied the instability criterion as well as it’s growth rate.

The equation of state for degenerate electrons in such intersteller compact objects are mathematically explained by Chandrasekhar for two limits, namely non-relativistic and ultrarelativistic limits [10,11]. The degenerate electron equation of state of Chandrasekhar is ${p}_{e}$ is proportional to ${N}_{e}^{5/3}$ for the non-relativistic limit and ${p}_{e}$ is proportional to ${N}_{e}^{4/3}$ the ultrarelativistic limit where ${p}_{e}$ is the degenerate electron pressure and ${N}_{e}$ is the degenerate electron number density. Mamun and Shukla considered an unmagnetized degenerate plasma without and with negatively charged stationary dust [12, 13]. Mamun et al, studied the ion acoustic Solitary Waves (SW’s) in the presence of an external magnetic field for ultra-relativistic degenerate electron-ion plasmas using the reductive perturbation technique [14]. Using the same technique Haider et al, also studied the obliquely propagating solitary structure with the presence of external magnetic field [15,16]. But the presence of heavy ions, which can be arbitrary charged, were not considered in the work of Mamun et al and Haider et al [14,15].

In the present work we have studied the nonlinear propagation of DA SW’s in unmagnetized and magnetized degenerate plasmas with adiabatic pressure. We have studied both ultra-relativistic and non-relativistic case simultaneously using the generalized equation proposed by Haider [17]. To do this we have derived Korteweg-de Vries (K-dV) and Zakharov-Kuznetsov (ZK) equation by reductive perturbation method and find out the solution of it [6]. We have studied the instability criterion as well as it’s growth rate.

Plasma Model

We have studied the nonlinear propagation of DASW’s in
unmagnetized and magnetized degenerate plasmas for adiabatic
situation containing,

(i) inertial positive mobile dust component

(ii) inertialess degenerate electrons

(iii) inertialess degenerate positrons

(iv) stationary negative dust component

(v) inertial Maxwellian ions

At equilibrium, we have, $${n}_{pd0}+{n}_{i0}+{n}_{p0}={n}_{nd0}+{n}_{e0}\text{(1)}$$ where, ${n}_{pd0},\text{}{n}_{i0},\text{\hspace{0.17em}}{n}_{p0},\text{\hspace{0.17em}}{n}_{nd0},\text{}{n}_{e0}$ are the positive dust, ion, positron, negative dust and electron number densities at equilibrium respectively.

(i) inertial positive mobile dust component

(ii) inertialess degenerate electrons

(iii) inertialess degenerate positrons

(iv) stationary negative dust component

(v) inertial Maxwellian ions

At equilibrium, we have, $${n}_{pd0}+{n}_{i0}+{n}_{p0}={n}_{nd0}+{n}_{e0}\text{(1)}$$ where, ${n}_{pd0},\text{}{n}_{i0},\text{\hspace{0.17em}}{n}_{p0},\text{\hspace{0.17em}}{n}_{nd0},\text{}{n}_{e0}$ are the positive dust, ion, positron, negative dust and electron number densities at equilibrium respectively.

Solitary Waves in Unmagnetized Plasmas

Basic Equations

The dynamics of such DASW’s in one dimensional form and is
given by the followings equations,
$$\frac{\partial {n}_{pd}}{\partial t}+\frac{\partial}{\partial x}({n}_{pd}{u}_{pd})=0\text{(2)}$$
$$\frac{\partial {u}_{pd}}{\partial t}+({u}_{pd}\frac{\partial}{\partial x}){u}_{pd}=-\frac{\partial \phi}{\partial x}-(\frac{{\sigma}_{d}}{{n}_{pd}}\text{)}\frac{\partial}{\partial x}{p}_{pd}\text{(3)}$$
$$\frac{\partial {u}_{pd}}{\partial t}+({u}_{pd}\frac{\partial}{\partial x}){p}_{pd}\text{+}\alpha {p}_{pd}\frac{\partial}{\partial x}{u}_{pd}\text{=0(4)}$$
$$\frac{{\partial}^{2}\phi}{\partial {x}^{2}}=({\mu}_{e}{n}_{e}-{\mu}_{p}{n}_{p}-{n}_{pd}-{\mu}_{i}{n}_{i}+{\mu}_{nd}{n}_{nd})\text{(5)}$$
where,
${n}_{pd}$
is the positive dust number density normalized by
its equilibrium value
${n}_{pd0},\text{}{n}_{nd}$
is the negative dust number density
normalized by its equilibrium value
${n}_{nd0},\text{}{n}_{p}$
is the positron
number density normalized by its equilibrium value
${n}_{p0},\text{}{n}_{i}$
is the
ion number density normalized by its equilibrium value
${n}_{i0},\text{}{n}_{e}$
is
the electron number density normalized by its equilibrium value
${n}_{e0},\text{}{u}_{pd}$
is the dust fluid speed normalized by
${C}_{pd}={({m}_{nd}{c}^{2}/{m}_{pd})}^{1/2},\text{}{m}_{pd}$
is the rest mass of positive dust,
${m}_{nd}$
is the mass of negative
dust and C being the speed of light.
$\phi $
is the DA electrostatic wave
potential normalized by
$\mathrm{(}{m}_{pd}{C}_{pd}^{2}\mathrm{/}{z}_{d}e\mathrm{)}$
, with
${z}_{d}$
is the number
of positive charge residing on the positive charged dust and e
being the magnitude of unit charge.
$\alpha $
is the adiabatic index.
${\sigma}_{d}$
is the ratio of the positive dust temperature to negative dust
temperature.
${\mu}_{e}\mathrm{=}{n}_{e0}\mathrm{/}{n}_{pd0},\text{}{\mu}_{p}\mathrm{=}{n}_{p0}\mathrm{/}{n}_{pd0}\text{,}{\mu}_{i}\mathrm{=}{n}_{i0}\mathrm{/}{n}_{pd0}$
and
${\mu}_{nd}\mathrm{=}{n}_{nd0}\mathrm{/}{n}_{pd0}$

Here the space variables are normalized by Debye radius ${\lambda}_{d}={({m}_{pd}{c}^{2}/4\pi {n}_{d0}{z}_{d}^{2}{e}^{2})}^{1/2}$ and time variable

Non-thermal distributed ions can be represented by, $${n}_{i}\mathrm{=(1}-\phi +\frac{1}{2}{\phi}^{2}\mathrm{)}\text{(6)}$$ According to Haider the value of ${n}_{e}$ and ${n}_{p}$ can be express respectively as [18, 19], $${n}_{e}={(1+\frac{\gamma -1}{\beta}\phi )}^{\frac{1}{\gamma -1}}\text{(7)}$$ $$and\text{}{n}_{p}={(1-\frac{\gamma -1}{\beta}\phi )}^{\frac{1}{\gamma -1}}\text{(8)}$$ where, $\beta \mathrm{=(}\frac{K}{E}\mathrm{)}{n}_{0}^{\mathrm{(}\gamma -\mathrm{1)}}$ , where, $K\mathrm{=}\frac{1}{\mathrm{18(}\gamma -\mathrm{1)}}\text{}\mathrm{(}\frac{3{\Delta}^{3}}{\pi}\mathrm{)}{\text{}}^{\mathrm{(}\gamma -\mathrm{1)}}\text{}E$ with $\Lambda \mathrm{(=}\frac{h}{{m}_{pd}c}\mathrm{)}$ is the Compton wavelength normalized by Debye radious $({\lambda}_{d}),\text{}E\mathrm{=}{m}_{pd}{c}^{2}$ and ${n}_{0}$ be the number density of plasma particle. $\gamma \mathrm{=4/3}$ for ultra-relativistic case and $\gamma \mathrm{=5/3}$ for nonrelativistic case.

Here the space variables are normalized by Debye radius ${\lambda}_{d}={({m}_{pd}{c}^{2}/4\pi {n}_{d0}{z}_{d}^{2}{e}^{2})}^{1/2}$ and time variable

*(t)*is normalized by ${\omega}_{pd}^{-1}={({m}_{pd}/4\pi {n}_{pd0}{z}_{d}^{2}{e}^{2})}^{1/2}$.Non-thermal distributed ions can be represented by, $${n}_{i}\mathrm{=(1}-\phi +\frac{1}{2}{\phi}^{2}\mathrm{)}\text{(6)}$$ According to Haider the value of ${n}_{e}$ and ${n}_{p}$ can be express respectively as [18, 19], $${n}_{e}={(1+\frac{\gamma -1}{\beta}\phi )}^{\frac{1}{\gamma -1}}\text{(7)}$$ $$and\text{}{n}_{p}={(1-\frac{\gamma -1}{\beta}\phi )}^{\frac{1}{\gamma -1}}\text{(8)}$$ where, $\beta \mathrm{=(}\frac{K}{E}\mathrm{)}{n}_{0}^{\mathrm{(}\gamma -\mathrm{1)}}$ , where, $K\mathrm{=}\frac{1}{\mathrm{18(}\gamma -\mathrm{1)}}\text{}\mathrm{(}\frac{3{\Delta}^{3}}{\pi}\mathrm{)}{\text{}}^{\mathrm{(}\gamma -\mathrm{1)}}\text{}E$ with $\Lambda \mathrm{(=}\frac{h}{{m}_{pd}c}\mathrm{)}$ is the Compton wavelength normalized by Debye radious $({\lambda}_{d}),\text{}E\mathrm{=}{m}_{pd}{c}^{2}$ and ${n}_{0}$ be the number density of plasma particle. $\gamma \mathrm{=4/3}$ for ultra-relativistic case and $\gamma \mathrm{=5/3}$ for nonrelativistic case.

K-dV Equation

We now follow the reductive perturbation method and
construct a weakly nonlinear theory for the DA waves with small
but finite amplitude, which leads to a scaling of the independent
variables through the stretched coordinates as [6, 18, 19],

$$Z\mathrm{=}{\epsilon}^{\mathrm{1/2}}\text{}\mathrm{(}x-{v}_{p}t\mathrm{)}\text{(9)}$$
$$\tau \mathrm{=}{\epsilon}^{\mathrm{3/2}}\text{}t\text{(10)}$$
where,
$\epsilon $
is a smallness parameter measuring the weakness
of the dispersion,
${v}_{p}$
is the nonlinear wave phase velocity. We can
expand the perturbed quantities
${n}_{pd},\text{}{u}_{pd},\text{}{p}_{pd}$
and
$\phi $
bout
their equilibrium values in powers of
$\epsilon $,

$${n}_{pd}\mathrm{=1}+\epsilon {n}_{pd}^{\mathrm{(1)}}+{\epsilon}^{2}{n}_{pd}^{\mathrm{(2)}}+\mathrm{...}\text{(11)}$$
$${u}_{pd}\mathrm{=0}+\epsilon {u}_{pd}^{\mathrm{(1)}}+{\epsilon}^{2}{u}_{pd}^{\mathrm{(2)}}+\mathrm{...}\text{(12)}$$
$$\phi \mathrm{=0}+\epsilon {\phi}^{\mathrm{(1)}}+{\epsilon}^{2}{\phi}^{\mathrm{(2)}}+\mathrm{...}\text{(13)}$$
$${p}_{pd}\mathrm{=1}+\epsilon {p}_{pd}^{\mathrm{(1)}}+{\epsilon}^{2}{p}_{pd}^{\mathrm{(2)}}+\mathrm{...}\text{(14)}$$
Using the stretched coordinates and equations (11)-(14) in
equations (2)-(5); and equating the coefficients of
${\epsilon}^{\mathrm{3/2}}$
from the
continuity and momentum equation and coefficients of
$\epsilon $
from
Poisson’s equation and rearranging the parameters one can
obtain the linear dispersion relation for the DASW’s

$${v}_{p}\mathrm{=}\sqrt{\frac{1}{\mathrm{(}\frac{1}{\beta}{\mu}_{e}+\frac{1}{\beta}{\mu}_{p}+{\mu}_{i}\mathrm{)}}+\alpha {\sigma}_{d}}\text{(15)}$$
It can be seen from equation (15) that the presence of the
Maxwellian ions significantly modifies the linear dispersion
relation.

Equating the next higher order co-efficient of from above equations and using the parameters we can finally obtain a K-dV equation describing the nonlinear propagation of the DASW’s in the dusty plasma

Equating the next higher order co-efficient of from above equations and using the parameters we can finally obtain a K-dV equation describing the nonlinear propagation of the DASW’s in the dusty plasma

**Figure 1:**

*A=0*graph, showing the variation of $\beta $ with ${\mu}_{e}$ and ${\mu}_{p}$ for the values of $\alpha \mathrm{=0.5},\text{}{\mu}_{i}\mathrm{=0.4}$ and $\gamma \mathrm{=5/3}$

**Figure 2:**

*A=0*graph, showing the variation of $\beta $ with ${\mu}_{e}$ and ${\mu}_{p}$ for the values of $\alpha \mathrm{=0.5},\text{}{\mu}_{i}\mathrm{=0.4}$ and $\gamma \mathrm{=4/3}$

where
$$A\mathrm{=}\left[{\mu}_{i}-\mathrm{(}{\mu}_{e}-{\mu}_{p}\mathrm{)(}\frac{2-\gamma}{{\beta}^{2}}\mathrm{)}\right]\frac{{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}{2{v}_{p}}\text{(17)}$$
$$B\mathrm{=}\frac{{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}{2{v}_{p}}\text{(18)}$$

Solution of K-dV Equation

The stationary solution of this K-dV equation can be obtained
by transforming the independent variables
$\xi $
and
$\tau $
to
$\xi \mathrm{=}Z-{u}_{0}\tau ,\text{}\tau \mathrm{=}t$
where
${u}_{0}$
is a constant solitary wave velocity. For simplicity,
we have write
${\phi}^{\mathrm{(1)}}$
as
$\phi $. Now using the appropriate boundary
conditions for localized disturbances, viz.
$${\phi}^{\mathrm{(1)}}\to 0,\text{}\mathrm{(}d{\phi}^{\mathrm{(1)}}\text{}\mathrm{/}d\xi \mathrm{)}\to 0,\text{}\mathrm{(}{d}^{2}{\phi}^{\mathrm{(1)}}\text{}\mathrm{/}d{\xi}^{2}\mathrm{)}\to 0$$
at $\xi \to \pm \infty $
. Thus, one can express the stationary solution of
this K-dV equation as
$$\phi \mathrm{=}{\phi}_{m}sec{h}^{2}\mathrm{[}\xi \mathrm{/}\Delta \mathrm{]}\text{(19)}$$
where
${\phi}_{m}\mathrm{=(3}{u}_{0}\mathrm{/}A\mathrm{)}$
is the amplitude and
$\Delta \mathrm{=}\sqrt{\frac{4B}{{u}_{0}}}$
is the width
of the solitary waves. The value of A and B are shown in (17)
and (18) respectively.

**Figure 3:**Variation of $\phi $ with $\xi $ for the values of $\alpha \mathrm{=0.5},$ ${\mu}_{p}\mathrm{=1},$ ${\mu}_{i}\mathrm{=0.4},\text{}{u}_{0}\mathrm{=0.1},$ ${\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=1},$ $\beta \mathrm{=1},\text{}\gamma \mathrm{=5/3}$ (solid curve) and $\gamma \mathrm{=4/3}$ (dotted curve).

**Figure 4:**Variation of $\phi $ with $\xi $ for the values of $\alpha \mathrm{=0.5},\text{}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{\mu}_{e}\mathrm{=1.2},\text{}{u}_{0}\mathrm{=0.1},\text{}{\sigma}_{d}\mathrm{=1},\text{}\beta \mathrm{=0.8}$ (Black), $\beta \mathrm{=0.9}$ (Blue) and $\beta \mathrm{=1}$ (Red). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

Solitary Waves in Magnetized Plasmas

Basic Equations

We have been considered that, there an external static
magnetic field present ${B}_{0}$
acting along the z-direction
$({B}_{0}\mathrm{=}\widehat{k}{B}_{0}\mathrm{)}$
where
$\widehat{k}$
is the unit vector along the z-direction, so that the ions
and dusts are moving along the magnetic field direction.

The dynamics of such DASW’s in three dimensional form and is given by the followings equations, $$\frac{\partial {n}_{pd}}{\partial t}+\nabla \mathrm{.(}{n}_{pd}{u}_{pd}\mathrm{)=0}\text{(20)}$$ $$\begin{array}{l}\frac{\partial {u}_{pd}}{\partial t}+\mathrm{(}{u}_{pd}\mathrm{.}\nabla \mathrm{)}{u}_{pd}\mathrm{=}-\nabla \mathrm{.}\phi +{\omega}_{cd}\mathrm{(}{u}_{pd}\times \widehat{k}\mathrm{)}\\ \text{\hspace{1em}}\text{\hspace{1em}}-\mathrm{(}\frac{{\sigma}_{d}}{{n}_{pd}}\mathrm{)}\nabla {p}_{pd}\text{(21)}\end{array}$$ $$\frac{\partial {p}_{pd}}{\partial t}+\mathrm{(}{u}_{pd}\mathrm{.}\nabla \mathrm{)}{p}_{pd}+\alpha {p}_{pd}\nabla \mathrm{.}{u}_{pd}\mathrm{=0}\text{(22)}$$ $${\nabla}^{2}\phi \mathrm{=(}{\mu}_{e}{n}_{e}-{\mu}_{p}{n}_{p}-{n}_{pd}-{\mu}_{i}{n}_{i}+{\mu}_{nd}\text{}{n}_{nd}\mathrm{)}\text{(23)}$$ Here, ${\omega}_{cd}$ is the positive dust cyclotron frequency $\mathrm{(}{z}_{d}e{B}_{0}\mathrm{/}{m}_{pd0}c\mathrm{)}$ normalized by $${\omega}_{pd}\mathrm{=(}\frac{4\pi {n}_{pd0}\text{}{z}_{d}^{2}\text{}{e}^{2}}{{m}_{pd}}\mathrm{)}{\text{}}^{\frac{1}{2}}$$

The dynamics of such DASW’s in three dimensional form and is given by the followings equations, $$\frac{\partial {n}_{pd}}{\partial t}+\nabla \mathrm{.(}{n}_{pd}{u}_{pd}\mathrm{)=0}\text{(20)}$$ $$\begin{array}{l}\frac{\partial {u}_{pd}}{\partial t}+\mathrm{(}{u}_{pd}\mathrm{.}\nabla \mathrm{)}{u}_{pd}\mathrm{=}-\nabla \mathrm{.}\phi +{\omega}_{cd}\mathrm{(}{u}_{pd}\times \widehat{k}\mathrm{)}\\ \text{\hspace{1em}}\text{\hspace{1em}}-\mathrm{(}\frac{{\sigma}_{d}}{{n}_{pd}}\mathrm{)}\nabla {p}_{pd}\text{(21)}\end{array}$$ $$\frac{\partial {p}_{pd}}{\partial t}+\mathrm{(}{u}_{pd}\mathrm{.}\nabla \mathrm{)}{p}_{pd}+\alpha {p}_{pd}\nabla \mathrm{.}{u}_{pd}\mathrm{=0}\text{(22)}$$ $${\nabla}^{2}\phi \mathrm{=(}{\mu}_{e}{n}_{e}-{\mu}_{p}{n}_{p}-{n}_{pd}-{\mu}_{i}{n}_{i}+{\mu}_{nd}\text{}{n}_{nd}\mathrm{)}\text{(23)}$$ Here, ${\omega}_{cd}$ is the positive dust cyclotron frequency $\mathrm{(}{z}_{d}e{B}_{0}\mathrm{/}{m}_{pd0}c\mathrm{)}$ normalized by $${\omega}_{pd}\mathrm{=(}\frac{4\pi {n}_{pd0}\text{}{z}_{d}^{2}\text{}{e}^{2}}{{m}_{pd}}\mathrm{)}{\text{}}^{\frac{1}{2}}$$

**Figure 5:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=1},\text{}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{\mu}_{e}\mathrm{=1.2},\text{}{u}_{0}\mathrm{=0.1},\text{}{\sigma}_{d}\mathrm{=1},\text{}\alpha \mathrm{=0.3}$ (Black), $\alpha \mathrm{=0.4}$ (Red) and $\alpha \mathrm{=0.5}$ (Blue). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$

**Figure 6:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=1},\text{}\alpha \mathrm{=0.5},\text{}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{u}_{0}\mathrm{=0.1},\text{}{\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=0.6}$ (Black), ${\sigma}_{d}\mathrm{=0.8}$ (Red) and ${\sigma}_{d}\mathrm{=1}$ (Blue). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

Zakharov-Kuznetsov Equation

To derive the Zakharov-Kuznetsov equation, which is known
as `K-dV equation in three dimensions’, we first introduce the
following stretched coordinates
$$X\mathrm{=}{\epsilon}^{\mathrm{1/2}}\text{}x\text{(24)}$$
$$Y\mathrm{=}{\epsilon}^{\mathrm{1/2}}\text{}y\text{(25)}$$
$$Z\mathrm{=}{\epsilon}^{\mathrm{1/2}}\text{}\mathrm{(}z-{v}_{p}t\mathrm{)}\text{(26)}$$
$$\tau \mathrm{=}{\epsilon}^{\mathrm{3/2}}\text{}t\text{(27)}$$
We can expand the perturbed quantities
${n}_{pd},\text{}{u}_{pdx},\text{}{u}_{pdy},\text{}{u}_{pdz},\text{}{p}_{pd}$
and $\phi $
about their equilibrium values in
powers of $\epsilon $

$${n}_{pd}\mathrm{=1}+\epsilon {n}_{pd}^{\mathrm{(1)}}+{\epsilon}^{2}{n}_{pd}^{\mathrm{(2)}}+\mathrm{...}\text{(28)}$$
$${u}_{pdx}\mathrm{=0}+{\epsilon}^{\mathrm{3/2}}{u}_{pdx}^{\mathrm{(1)}}+{\epsilon}^{2}{u}_{pdx}^{\mathrm{(2)}}+\mathrm{...}\text{(29)}$$
$${u}_{pdy}\mathrm{=0}+{\epsilon}^{\mathrm{3/2}}{u}_{pdy}^{\mathrm{(1)}}+{\epsilon}^{2}{u}_{pdy}^{\mathrm{(2)}}+\mathrm{...}\text{(30)}$$
$${u}_{pdz}\mathrm{=0}+\epsilon {u}_{pdz}^{\mathrm{(1)}}+{\epsilon}^{2}{u}_{pdz}^{\mathrm{(2)}}+\mathrm{...}\text{(31)}$$
$$\phi \mathrm{=0}+\epsilon {\phi}^{\mathrm{(1)}}+{\epsilon}^{2}{\phi}^{\mathrm{(2)}}+\mathrm{...}\text{(32)}$$
$${p}_{pd}\mathrm{=1}+\epsilon {p}_{pd}^{\mathrm{(1)}}+{\epsilon}^{2}{p}_{pd}^{\mathrm{(2)}}+\mathrm{...}\text{(33)}$$
**Figure 7:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=1},\text{}\alpha \mathrm{=0.5},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{u}_{0}\mathrm{=0.1},\text{}{\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=1},\text{}{\mu}_{p}\mathrm{=0.8}$ (Blue), ${\mu}_{p}\mathrm{=0.9}$ (Black) and ${\mu}_{p}\mathrm{=1}$ (Red). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

Next, substituting eqs. (24)-(33) into eqs. (20)-(23) and
obtain the lowest order equations of (20)-(23) which in turn
can be solved as (15), the linear dispersion relation for DASW’s.
This implies that linear dispersion relation for unmagnetized and
magnetized situations are same. Magnetic field does not effect on
the linear dispersion relation.

Equating the next higher order co-efficient of $\epsilon $ from above equations and by the use of parameters we can finally obtained ZK equation describing the nonlinear propagation of the DA SW’s in the dusty plasma $$\text{\hspace{1em}}\text{\hspace{1em}}\frac{\partial {\phi}^{\mathrm{(1)}}}{\partial \tau}+PQ{\phi}^{\mathrm{(1)}}\text{}\frac{\partial {\phi}^{\mathrm{(1)}}}{\partial Z}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}+\frac{1}{2}P\frac{\partial}{\partial Z}\left[\frac{{\partial}^{2}}{\partial {Z}^{2}}+R\left(\frac{{\partial}^{2}}{\partial {X}^{2}}+\frac{{\partial}^{2}}{\partial {Y}^{2}}\right)\right]{\phi}^{\mathrm{(1)}}\text{}\mathrm{=0}\text{(34)}$$ where, $$\text{\hspace{1em}}\text{\hspace{1em}}P\mathrm{=}\frac{{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}{{v}_{p}}\text{(35)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}Q\mathrm{=}\frac{1}{2}\left[{\mu}_{i}-\mathrm{(}{\mu}_{e}-{\mu}_{p}\mathrm{)(}\frac{2-\gamma}{{\beta}^{2}}\mathrm{)}\right]\text{(36)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}R\mathrm{=1}+\frac{{v}_{p}^{4}}{{\omega}_{cd}^{2}{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}\text{(37)}$$ Equation (34) is a ZK equation for adiabatic DASW’s with Maxwellian ions, degenerate electrons and positrons.

Equating the next higher order co-efficient of $\epsilon $ from above equations and by the use of parameters we can finally obtained ZK equation describing the nonlinear propagation of the DA SW’s in the dusty plasma $$\text{\hspace{1em}}\text{\hspace{1em}}\frac{\partial {\phi}^{\mathrm{(1)}}}{\partial \tau}+PQ{\phi}^{\mathrm{(1)}}\text{}\frac{\partial {\phi}^{\mathrm{(1)}}}{\partial Z}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}+\frac{1}{2}P\frac{\partial}{\partial Z}\left[\frac{{\partial}^{2}}{\partial {Z}^{2}}+R\left(\frac{{\partial}^{2}}{\partial {X}^{2}}+\frac{{\partial}^{2}}{\partial {Y}^{2}}\right)\right]{\phi}^{\mathrm{(1)}}\text{}\mathrm{=0}\text{(34)}$$ where, $$\text{\hspace{1em}}\text{\hspace{1em}}P\mathrm{=}\frac{{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}{{v}_{p}}\text{(35)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}Q\mathrm{=}\frac{1}{2}\left[{\mu}_{i}-\mathrm{(}{\mu}_{e}-{\mu}_{p}\mathrm{)(}\frac{2-\gamma}{{\beta}^{2}}\mathrm{)}\right]\text{(36)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}R\mathrm{=1}+\frac{{v}_{p}^{4}}{{\omega}_{cd}^{2}{\mathrm{(}{v}_{p}^{2}-\alpha {\sigma}_{d}\mathrm{)}}^{2}}\text{(37)}$$ Equation (34) is a ZK equation for adiabatic DASW’s with Maxwellian ions, degenerate electrons and positrons.

**Figure 8:**$P\mathrm{=0}$ graph, showing the variation of $\beta $ with ${\mu}_{e}$ and ${\mu}_{p}$ for the values of $\alpha \mathrm{=0.5},\text{}{\mu}_{i}\mathrm{=0.4}$ and $\gamma \mathrm{=5/3}$

**Figure 9:**$P\mathrm{=0}$ graph, showing the variation of $\beta $ with ${\mu}_{e}$ and ${\mu}_{p}$ for the values of $\alpha \mathrm{=0.5},\text{}{\mu}_{i}\mathrm{=0.4}$ and $\gamma \mathrm{=4/3}$

**Figure 10:**Variation of $\phi $ with $\xi $ for the values of ${\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{\mu}_{e}\mathrm{=1.2},$ ${\sigma}_{d}\mathrm{=1},\text{}{\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{}}^{0},$ $\beta \mathrm{=0.7},\text{}{u}_{0}\mathrm{=0.1},\text{}\alpha \mathrm{=0.3}$ (Red), $\alpha \mathrm{=0.4}$ (Blue) and $\alpha \mathrm{=0.5}$ (Black). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$

**Figure 11:**Variation of $\phi $ with $\xi $ for the values of $\alpha \mathrm{=0.5},\text{}\beta \mathrm{=0.7},\text{}{u}_{0}\mathrm{=0.1}$ ${\mu}_{p}\mathrm{=1},\text{}{\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{}}^{0},$ ${\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4}$ (Black), ${\mu}_{i}\mathrm{=0.5}$ (Blue) and ${\mu}_{i}\mathrm{=0.6}$ (Red). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

Solution of Zakharov-Kuznetsov Equation

The stationary solution (for simplicity, we have write
${\phi}^{\mathrm{(1)}}$
as $\phi $
) of this ZK equation as,

$$\text{\hspace{1em}}\text{\hspace{1em}}\phi \mathrm{(}\xi \mathrm{)=}{\phi}_{m}sec{h}^{2}\mathrm{(}k\xi \mathrm{)}\text{(38)}$$
where
${\phi}_{m}\mathrm{=[}\frac{3{u}_{0}}{{\delta}_{1}}\mathrm{]}$
is the amplitude and
$k\mathrm{=}\sqrt{\frac{{u}_{0}}{4{\delta}_{2}}}$
is the inverse
of the width ($\Delta $) of the SW’s respectively; with
${\delta}_{1}\mathrm{=}PQcos\delta $
and
${\delta}_{2}\mathrm{=}\frac{1}{2}P\mathrm{(}{{\displaystyle cos}}^{3}\delta +R{{\displaystyle sin}}^{2}\delta cos\delta \mathrm{)},\text{}\delta $
being the propagation
angle of SW’s.

The $A\mathrm{=0(}P\mathrm{=0)}$ surface plots are shown for non-relativistic and ultra-relativistic case in absence (presence) of magnetic field are shown in figure 1 and 2 (8 and 9) respectively. From these figures we find that the SW’s have positive potentials above the surface and negative potentials below the surface. Figure 3 shows the variation of $\phi $ with $\xi $ for ultra-relativistic and non-relativistic cases. Figure 4, 5, 6 and 7 showing the variation of $\phi $ with $\xi $ respectively for the different value of $\alpha ,\text{}{\sigma}_{d},\text{}\beta $ and ${\mu}_{p}$. All of these figure shows the variation of amplitude and width for ultrarelativistic and non-relativistic cases in the absence of magnetic field. Figure 10, 11, 12, 13, 14 and 15 showing the variation of $\phi $ with $\xi $ for the value of $\alpha ,\text{}{\mu}_{i},\text{}\delta ,\text{}{\mu}_{e},\text{}{\sigma}_{d}$ and $\beta $ respectively in the presence of magnetic field.

The $A\mathrm{=0(}P\mathrm{=0)}$ surface plots are shown for non-relativistic and ultra-relativistic case in absence (presence) of magnetic field are shown in figure 1 and 2 (8 and 9) respectively. From these figures we find that the SW’s have positive potentials above the surface and negative potentials below the surface. Figure 3 shows the variation of $\phi $ with $\xi $ for ultra-relativistic and non-relativistic cases. Figure 4, 5, 6 and 7 showing the variation of $\phi $ with $\xi $ respectively for the different value of $\alpha ,\text{}{\sigma}_{d},\text{}\beta $ and ${\mu}_{p}$. All of these figure shows the variation of amplitude and width for ultrarelativistic and non-relativistic cases in the absence of magnetic field. Figure 10, 11, 12, 13, 14 and 15 showing the variation of $\phi $ with $\xi $ for the value of $\alpha ,\text{}{\mu}_{i},\text{}\delta ,\text{}{\mu}_{e},\text{}{\sigma}_{d}$ and $\beta $ respectively in the presence of magnetic field.

**Figure 12:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=0.7},\text{}\alpha \mathrm{=0.5},\text{}{u}_{0}\mathrm{=0.1},$ ${\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},\text{}{\mu}_{e}\mathrm{=1.2},$ ${\sigma}_{d}\mathrm{=1},\text{}{\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{}}^{0},$ (Red), $\delta \mathrm{=30}{\text{}}^{0}$ (Blue) and $\delta \mathrm{=40}{\text{}}^{0}$ (Black). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

Instability

We now study the instability of the opaquely propagating
solitary waves, discussed in the previous section, by the method
of small- perturbation expansion (22-28). We first assume that
separate
$\phi $
as a function of
${\psi}_{0}$
and $\psi $
as

$$\text{\hspace{1em}}\text{\hspace{1em}}\phi \mathrm{=}{\psi}_{0}\mathrm{(}Z\mathrm{)}+\psi \mathrm{(}\xi \mathrm{,}\zeta \mathrm{,}\eta \mathrm{,}t\mathrm{)}\text{(39)}$$
Where
$\xi ,\text{}\zeta ,\text{}\eta $
and $t$
are components of new reference frame.
For a long-wavelength plane wave perturbation in a direction
with direction cosines
$\mathrm{(}{l}_{\zeta}\mathrm{,}{l}_{\eta}\mathrm{,}{l}_{\xi}\mathrm{)},\text{}\psi $
is given by

$$\text{\hspace{1em}}\text{\hspace{1em}}\psi \mathrm{=}\varphi \mathrm{(}Z\mathrm{)}{e}^{i\mathrm{[}k\mathrm{(}{l}_{\zeta}\zeta +{l}_{\eta}\eta +{l}_{\xi}Z\mathrm{)}-\omega t\mathrm{]}}\text{(40)}$$
in which
${l}_{\zeta}^{2}+{l}_{\eta}^{2}+{l}_{\xi}^{2}\mathrm{=1},\text{}k$
is the wave constant,
$\omega $
is the angular
frequency of the waves and
$\varphi $
is the amplitude of the waves.

For small $k,\text{}\varphi \mathrm{(}Z\mathrm{)}$ and $\omega $ can be expanded as $$\text{\hspace{1em}}\text{\hspace{1em}}\varphi \mathrm{(}Z\mathrm{)=}{\varphi}_{0}\mathrm{(}Z\mathrm{)}+k{\varphi}_{1}\mathrm{(}Z\mathrm{)}+{k}^{2}{\varphi}_{2}\mathrm{(}Z\mathrm{)}+\cdot \cdot \cdot \text{(41)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}\omega \mathrm{=}k{\omega}_{1}+{k}^{2}{\omega}_{2}+\cdot \cdot \cdot \text{(42)}$$ Doing some mathematical analysis we arrive at the following dispersion relation:

For small $k,\text{}\varphi \mathrm{(}Z\mathrm{)}$ and $\omega $ can be expanded as $$\text{\hspace{1em}}\text{\hspace{1em}}\varphi \mathrm{(}Z\mathrm{)=}{\varphi}_{0}\mathrm{(}Z\mathrm{)}+k{\varphi}_{1}\mathrm{(}Z\mathrm{)}+{k}^{2}{\varphi}_{2}\mathrm{(}Z\mathrm{)}+\cdot \cdot \cdot \text{(41)}$$ $$\text{\hspace{1em}}\text{\hspace{1em}}\omega \mathrm{=}k{\omega}_{1}+{k}^{2}{\omega}_{2}+\cdot \cdot \cdot \text{(42)}$$ Doing some mathematical analysis we arrive at the following dispersion relation:

**Figure 13:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=0.7},\text{}{\sigma}_{d}\mathrm{=1},\text{}{u}_{0}\mathrm{=0.1},$ ${\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{}}^{0},\text{}\alpha \mathrm{=0.5}$ ${\mu}_{p}\mathrm{=1},\text{\hspace{0.17em}}{\mu}_{i}\mathrm{=0.4},\text{}{\mu}_{e}\mathrm{=1}$ (Red), ${\mu}_{e}\mathrm{=1.1}$ (Blue) and ${\mu}_{e}\mathrm{=1.2}$ (Black). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

**Figure 14:**Variation of $\phi $ with $\xi $ for the values of $\beta \mathrm{=0.7},\text{}{\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{\hspace{0.17em}}}^{0},$ $\alpha \mathrm{=0.5},\text{}{u}_{0}\mathrm{=0.1},\text{}{\mu}_{p}\mathrm{=1}$ ${\mu}_{i}\mathrm{=0.4},\text{\hspace{0.17em}}{\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=0.6}$ (Red), ${\sigma}_{d}\mathrm{=0.8}$ (Blue) and ${\sigma}_{d}\mathrm{=1}$ (Black). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

where
$$\text{\hspace{1em}}\text{\hspace{1em}}\Omega \mathrm{=}\frac{2}{3}\mathrm{(}{\psi}_{m}{\mu}_{1}-2{\mu}_{2}{\kappa}^{2}\mathrm{)}$$
$$\text{\hspace{1em}}\text{\hspace{1em}}\Upsilon \mathrm{=}\frac{16}{45}\mathrm{(}{\psi}_{m}^{2}{\mu}_{1}^{2}-3{\psi}_{m}{\mu}_{1}{\mu}_{2}{\kappa}^{2}-3{\mu}_{2}^{2}{\kappa}^{4}+12{\delta}_{2}{\mu}_{3}{\kappa}^{4}\mathrm{)}$$
with,
$$\begin{array}{l}\text{\hspace{1em}}\text{\hspace{1em}}{\mu}_{1}\mathrm{=}{\delta}_{1}{l}_{\xi}+{\delta}_{3}{l}_{\zeta}\\ \text{\hspace{1em}}\text{\hspace{1em}}{\mu}_{2}\mathrm{=3}{\delta}_{2}{l}_{\xi}+{\delta}_{5}{l}_{\zeta}\\ \text{\hspace{1em}}\text{\hspace{1em}}{\mu}_{3}\mathrm{=3}{\delta}_{2}{l}_{\xi}^{2}+2{\delta}_{5}{l}_{\zeta}{l}_{\xi}+{\delta}_{6}{l}_{\zeta}^{2}+{\delta}_{7}{l}_{\eta}^{2}\end{array}$$
It is clear from the dispersion relation (43) that there is always
instability if
$\mathrm{(}\Upsilon -{\Omega}^{2}\mathrm{)>0}$
We can express the instability criterion as
$$\text{\hspace{1em}}\text{\hspace{1em}}{S}_{i}\mathrm{>0}\text{(44)}$$
with
$$\text{\hspace{1em}}\text{\hspace{1em}}{S}_{i}\mathrm{=}{l}_{\eta}^{2}\mathrm{[}{\omega}_{cd}^{2}+{{\displaystyle sin}}^{2}\delta \mathrm{]}+{l}_{\zeta}^{2}\left[{\omega}_{cd}^{2}-\frac{5}{3}\mathrm{(}{\omega}_{cd}^{2}+\mathrm{1)}{{\displaystyle tan}}^{2}\delta \right]\text{(45)}$$
If this instability criterion
${S}_{i}\mathrm{>0}$
is satisfied, the growth rate
$\Gamma =(\Upsilon -{\Omega}^{2}){\text{}}^{1/2}$
of the unstable perturbation of these solitary waves
is given by

**Figure 15:**Variation of $\phi $ with $\xi $ for the values of $\alpha \mathrm{=0.5},\text{}{\omega}_{cd}\mathrm{=0.3},\text{}\delta \mathrm{=20}{\text{}}^{0},$ ${u}_{0}\mathrm{=0.1},\text{\hspace{0.17em}}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},$ ${\mu}_{e}\mathrm{=1.2},\text{}{\sigma}_{d}\mathrm{=1},\text{}\beta \mathrm{=0.6}$ (Black), $\beta \mathrm{=0.7}$ (Blue) and $\beta \mathrm{=0.8}$ (Red). Here dotted curve represented ultra-relativistic $(\gamma \mathrm{=4/3})$ case and solid curve represented non-relativistic $(\gamma \mathrm{=5/3})$ case.

**Figure 16:**${S}_{i}\mathrm{=0}$ surface plot showing the variation of ${\omega}_{cd}$ with ${l}_{\eta}$ and ${l}_{\xi}$ for the values of ${\mu}_{e}\mathrm{=1.2},\text{}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},$ $\beta \mathrm{=1},\text{}\alpha \mathrm{=0.5},\text{}\sigma \mathrm{=1}$ and $\delta \mathrm{=20}{\text{}}^{0}$.

Figure 16 represent
${S}_{i}\mathrm{=0}$
surface plot showing the variation
of ${\omega}_{cd}$
with ${l}_{\eta}$
and ${l}_{\xi}$
, From this figure we found that
${\omega}_{cd}$
decreases
with increasing the value of
${l}_{\eta}$
very rapidly and increases slowly
with increasing the value of
${l}_{\xi}$
Figure 17 showing the variation of
growth rate
$\Gamma $
with ${\sigma}_{d}$
and $\beta $
We found from this figure that
$\Gamma $
decreases with increasing the value of
${\sigma}_{d}$
and $\beta $
Figure 18 shows
the variation of $\Gamma $
with ${\mu}_{i}$
and ${\mu}_{e}$
We have found from this figure
that
$\Gamma $
decreases with increasing the value of
${\mu}_{e}$
and ${\mu}_{i}$
Figure 19
has shown the variation of growth rate
$\Gamma $
with $\alpha $
and $\beta $
We have
seen from the figure that
$\Gamma $
increases with increasing the value of
$\alpha $.

Findings

DA SW’s have been examined in a collisionless dusty plasma
consisting of inertialess ultra-relativistic and non-relativistic
electrons and positrons, inertial mobile positive dust particles,
inertial Maxwellian ions by deriving K-dV equation using
reductive perturbation technique. It has been found that the basic
features of such DA solitary waves are significantly modified by
the presence degeneracy of adiabaticness of components. The
effects of the parametric regimes on solitary wave for both case
of unmagnetic and magnetic fields are discussed below:

1. Depending on the value of ${\mu}_{e},\text{}{\mu}_{p}$ and $\beta $ the solitary waves can be associated with positive (negative) potentials above (below) the A = 0 surface for unmagnetized and P = 0 surface for magnetized plasmas respectively.

2. The amplitude of the DASW’s is higher for ultrarelativistic case than non-relativistic one.

3. Magnetic field enhance the amplitude of the SW’s for both ultra-relativistic and non-relativistic case.

4. Both amplitude and width of the solitary waves increases with increasing the value of $\alpha ,\text{}\delta ,\text{}{\sigma}_{d}$ ${\mu}_{e}$ but decreases with increasing the value of ${\mu}_{i},\text{}{\mu}_{p}$ and $\beta $ in the case of unmagnetized as well as magnetized dusty plasmas.

5. Cyclotron frequency ${\omega}_{cd}$ increases with increasing the value of ${l}_{\eta}$ and decreasing the value of ${l}_{\xi}$.

6. The growth rate $\Gamma $ ecreases with increasing the value of $\sigma ,\text{}\beta ,\text{}{\mu}_{e}$ and ${\mu}_{i}$ but increases with increasing the value of $\alpha $.

1. Depending on the value of ${\mu}_{e},\text{}{\mu}_{p}$ and $\beta $ the solitary waves can be associated with positive (negative) potentials above (below) the A = 0 surface for unmagnetized and P = 0 surface for magnetized plasmas respectively.

2. The amplitude of the DASW’s is higher for ultrarelativistic case than non-relativistic one.

3. Magnetic field enhance the amplitude of the SW’s for both ultra-relativistic and non-relativistic case.

4. Both amplitude and width of the solitary waves increases with increasing the value of $\alpha ,\text{}\delta ,\text{}{\sigma}_{d}$ ${\mu}_{e}$ but decreases with increasing the value of ${\mu}_{i},\text{}{\mu}_{p}$ and $\beta $ in the case of unmagnetized as well as magnetized dusty plasmas.

5. Cyclotron frequency ${\omega}_{cd}$ increases with increasing the value of ${l}_{\eta}$ and decreasing the value of ${l}_{\xi}$.

6. The growth rate $\Gamma $ ecreases with increasing the value of $\sigma ,\text{}\beta ,\text{}{\mu}_{e}$ and ${\mu}_{i}$ but increases with increasing the value of $\alpha $.

**Figure 17:**Variation of growth rate ($\Gamma $) with $\sigma $ and $\beta $ for the values of ${\mu}_{e}\mathrm{=1.2},\text{\hspace{0.17em}}\text{}{\mu}_{p}\mathrm{=1},\text{}{\mu}_{i}\mathrm{=0.4},$ $\delta \mathrm{=2}{\text{}}^{0},\text{}\alpha \mathrm{=0.2},\text{\hspace{0.17em}}\text{}{l}_{\eta}\mathrm{=0.4},$ ${l}_{\xi}\mathrm{=0.5},\text{}{u}_{0}\mathrm{=0.1}$ and ${\omega}_{cd}\mathrm{=0.3}$.

**Figure 18:**Variation of growth rate ($\Gamma $ with ${\mu}_{i}$ and ${\mu}_{e}$ for the values of $\alpha \mathrm{=0.1},\text{}\beta \mathrm{=0.7},\text{}{\mu}_{p}\mathrm{=1},$ ${\omega}_{cd}\mathrm{=0.5},\text{}{l}_{\xi}\mathrm{=0.5},\text{}{l}_{\eta}\mathrm{=0.4},$ $\delta \mathrm{=15}{\text{}}^{0},\text{}\sigma \mathrm{=1}$ and ${u}_{0}\mathrm{=0.1}$.

**Figure 19:**Variation of growth rate ($\Gamma $) with ${l}_{\eta}$ and ${l}_{\xi}$ for the values of $\beta \mathrm{=0.7},\text{\hspace{0.17em}}\text{}{\mu}_{p}\mathrm{=1},\text{}{\omega}_{cd}\mathrm{=0.3},$ ${\mu}_{e}\mathrm{=1.2},\text{}{\mu}_{i}\mathrm{=0.4},\text{}\delta \mathrm{=2}{\text{}}^{0},$ $\sigma \mathrm{=1},\text{}{l}_{\xi}\mathrm{=0.5}$ and ${u}_{0}\mathrm{=0.1}$.

Conclusion

In conclusion we can say that Maxwellian ions can modify the
basic properties of the solitary waves significantly. On the other
hand the presence of magnetic field effects the amplitude and
width of the solitary wave and its instability. The properties of
the solitary waves also moderated by the presence the adiabetic
pressers in degenerate plasmas. We have shown the variation
of solitary wave phenomenon and its stabilities criterion as
well as its growth rate due to the ultra-relativistic and nonrelativistic
degenerate pressure. These theoretical analysis may
helpful astrophysicist to understand different critical situations
exist in white dwarf and neutron stars; not only that, the other
astrophysical objects where degenerate pressure plays a vital
role to sustain these. Compering the present analysis to others
(non-degenerate objects) one give a prediction on the situation of
degenerate astrophysical objects.

The plasma parameters used in the present investigations correspond to white dwarfs though it can also be applied for neutron stars. As the number density of plasma parameters, so as the value of $\beta $ , is much higher for neutron stars than white dwarfs; it can be predict that the amplitude of the SW’s would be higher. But numerical analysis is essential to view the true fact. We hope we will see the actual variation of wave properties between white dwarf and neutron stars very soon. We hope that our present investigation can helpful for understanding the white dwarfs and neutron stars and give a guideline for then who wants to work in the relevant field.

The plasma parameters used in the present investigations correspond to white dwarfs though it can also be applied for neutron stars. As the number density of plasma parameters, so as the value of $\beta $ , is much higher for neutron stars than white dwarfs; it can be predict that the amplitude of the SW’s would be higher. But numerical analysis is essential to view the true fact. We hope we will see the actual variation of wave properties between white dwarf and neutron stars very soon. We hope that our present investigation can helpful for understanding the white dwarfs and neutron stars and give a guideline for then who wants to work in the relevant field.

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