# Revisiting the Physics of Strong Coupling on Dusty Plasma System

J. Goswami and S. S. Kausik  Centre of Plasma Physics, Institute for Plasma
Research, Nazirakhat, Sonapur-782402, Kamrup (M), Assam, India
<EMAIL_ADDRESS>
###### Abstract

Dusty plasma can exist in a wide range, from the laboratory to Saturn’s rings.
The coupling parameter plays a very important role in diagnosing dusty plasma.
There are two ways to deal analytically with the effect of coupling in plasma.
First, where the coupling parameter is represented with the help of
viscoelastic relaxation time $(\tau_{m})$, and the second where the
electrostatic temperature $(d)$ represents the coupling parameters, in this
work, we have used both techniques in a strongly coupled four-component plasma
in which we have considered the dynamics of negatively charged dust grains and
$q-$ non-extensive distribution for electrons, ions, and positrons. The main
focus of this particular work is to find out the operational regions and
validity of these two techniques. The result of this paper will give future
researchers an idea of how coupling would be encountered in a dusty plasma
system.

  * November 2023

##  1 Introduction:

Dust grains in any plasma increase the complexity of the plasma, which is why
dusty plasma is most popularly known as ‘complex plasma’[1]. Dusty or complex
plasma can present in various astrophysical bodies [2, 3, 4, 5, 6, 7], namely,
in the planetary rings, in interstellar molecular clouds, in cometary comae
and tails, etc. Dust particles are also very much present in different types
of nebulae. On the other hand, dusty or complex plasma environments can also
be created in the laboratory. Polar mesosphere summer echoes (PMSE) can also
be explained by the presence of charged dust particles there[1, 8, 9]. There
are various physical processes like photoelectron emission, secondary electron
emission, thermionic emission, etc., by which dust grains can be negatively or
positively charged[10, 11, 12, 13, 14, 15].

Generally, astrophysical plasmas in Earth’s bow shock[16], the vicinity of the
moon [17], Jupiter’s magnetosphere[18], Saturn’s magnetosphere[19, 20], galaxy
clusters [21] , etc exhibit the long-range interactions. Therefore, these
systems do not obey the extensive statistics but follow the statistics
theoretically proposed by Tsallis[22]. The distribution following the above-
mentioned statistics is also observed by the Voyager I space mission [23].

Electron–positron–ion-dust (EPID) plasma has been widely found in various
astrophysical bodies like Jupiter’s magnetosphere[18], supernova
environments[24], Saturn’s magnetosphere[19, 20], the solar atmosphere[25,
26], galactic centre [27], and cometary tails[25, 28, 29]. There are three
main mechanisms viz. pair production[26], thermal heating[30, 31], and
radiative heating [32] by which charged dust species or positron or both occur
naturally in electron–ion plasma. The light positron and the heavy dust grains
very prominently control the dynamics of the EPID plasma medium (EPIDPM).
Electrostatic potential structure[33], the existence of the positive super
solitons [34], and the stability of the system [35] have been discussed by
various authors in this type of four-component plasma.

In this work, we are mainly focusing on the effect of coupling in the dusty
plasma. Dusty plasma can be divided into two varieties in light of coupling,
weakly coupled and strongly coupled dusty plasma. Theoretically, this
difference can be depicted by the coulomb coupling parameter,

$\Gamma=\frac{Q^{2}}{dT_{d}}exp(-\kappa)$ (1)

Where $Q=Z_{d0}e$; is the charge on the surface of the dust grains, $T_{d}$ is
the temperature of dust, and $\kappa=\frac{d}{\lambda_{d}}$; $d$ is an average
intergrain distance and $\lambda_{d}$ is the dusty plasma Debye radius. Thus,
$\Gamma\leq 1$ represents a weakly coupled plasma and $\Gamma>1$ represents a
strongly coupled plasma. Again, if $\Gamma>\Gamma_{c}$ (a critical value of
the coupling parameter) the dust grains form a crystalline structure. In the
general hydrodynamic model of dusty plasma, there are two ways to encounter
the problem of strong coupling,

a) when the coupling is represented by normalized viscoelastic relaxation time
[36, 37]

b) when the coupling is represented by the normalized electrostatic
temperature [38].

Various authors have used these two methodologies to handle the coupling
individually, but no one has ever done a comparative discussion of these two.
So, in this article, we basically want to address the following questions for
the first time:

a) Are there any connections between these two types of operations for strong
coupling?

b) If the answer to the previous question is no then why? But if the answer is
yes then what are the connections?

c) and lastly can we use any of these two procedures for all the plasma
systems? Are there any advantages or disadvantages of these models?

Here, we mainly focus on the four component dusty plasma present in Saturn’s
magnetosphere. But for simplicity, we have not taken the effect of the
magnetic field ( of strength $\sim 0.025-0.21G$ [39]) otherwise it will become
a 3D problem.[40].

There are two types of dissipation in the plasma system, collision, and
viscosity[41]. These dissipative forces come to act in the momentum equations.
Three situations can arise with the presence of dissipation. If the
dissipation is moderate then we have oscillatory shock waves. Monotonic shock
waves can be found with a higher value of dissipation, and if there is no
dissipation present in the system then solitary waves will be propagated in
the system. But, in an astrophysical scenario, the viscous co-efficient will
have a dominant effect over the collisional term [42].

In summary, we can say that we are focusing on viscous collisionless
astrophysical EPID plasma where we discuss for the first time two procedures
to deal with the strong coupling in the same plasma model. From these
calculations, we are able to conclude for the maiden time which operation is
suitable for analyzing the linear and non-linear characteristics of dust
acoustic (DA) waves. As we are working on dust acoustic (DA) mode, only the
motion of dust is considered to construct the model and the rest of the
quantities that is ions, electrons, and positrons have been $q-$nonextensive
distributions.

The paper is arranged in the following way- in section 2 we have formulated
the problem considering strongly coupled dust and this coupling is represented
by normalized viscoelastic relaxation time. Then we deal with the linear and
nonlinear characteristics of the same model in section 3. In section 4, we
have analyzed the model for different coupling conditions. Then section 5 we
have discussed some of the important results. Lastly, in section 6 we have
concluded with important findings from the work and the prospect.

##  2 Basic Formulations:

We have considered a four-component electron-ion-positron-dust plasma. To
investigate the dust acoustic mode we have to consider the normalized
dynamical equation of dust only. The continuity equation for the dust grains
is given by,

$\frac{\partial n_{d}}{\partial t}+\frac{\partial}{\partial
x}\left(n_{d}u_{d}\right)=0$ (2)

The momentum equation for strongly coupled dust is given by,

$\left(1+\tau_{m}D_{t}\right)\left[n_{d}\left(D_{t}u_{d}-\frac{\partial\varphi}{\partial
x}\right)\right]=\eta_{d}\frac{\partial^{2}u_{d}}{\partial x^{2}}$ (3)

Poisson’s equation bounds these two equations:

$\frac{\partial^{2}\varphi}{\partial
x^{2}}=n_{d}+\alpha_{e}n_{e}-\alpha_{i}n_{i}-\alpha_{p}n_{p}$ (4)

These equations (2-4) are normalized by standard normalization scheme and
these are, $\hat{x}\rightarrow\frac{x}{\lambda_{D}}$,
$\hat{t}\rightarrow\omega_{p}t$,
$\hat{\varphi}\rightarrow\frac{e\varphi}{T_{d}}$,
$\hat{u_{d}}\rightarrow\frac{u_{d}}{C_{d}}$, where,
$C_{d}=\left(\frac{Z_{d}T_{d}}{m_{d}}\right)^{1/2}$,
${\omega_{p}}^{-1}=\left(\frac{m_{d}}{4\pi Z_{d}e^{2}n_{d0}}\right)^{1/2}$ and
$\lambda_{D}=\left(\frac{T_{d}}{4\pi Z_{d}n_{d0}e^{2}}\right)^{1/2}$. Again,
$\eta_{d}$ is the normalized longitudinal viscous coefficient, and $\tau_{m}$
is the normalized viscoelastic relaxation time normalized by dust time period
$(\tau_{d})$ [36, 37].

We consider the $q-$nonextensive distribution for electrons, positrons, and
ions. The normalized number densities of these are given below,

$n_{e}=\left[1+\left(q_{e}-1\right)\sigma_{e}\varphi\right]^{\frac{1+q_{e}}{2\left(q_{e}-1\right)}}$
(5)
$n_{p}=\left[1-\left(q_{p}-1\right)\sigma_{p}\varphi\right]^{\frac{1+q_{p}}{2\left(q_{p}-1\right)}}$
(6)
$n_{i}=\left[1-\left(q_{i}-1\right)\sigma_{i}\varphi\right]^{\frac{1+q_{i}}{2\left(q_{i}-1\right)}}$
(7)

where, $\sigma_{j}=\frac{T_{j}}{T_{d}}$; $(j=e.p,i)$.

All the species have different $q-$values in the model. Basically, the degree
of non-extensivity is measured by this non-extensive parameter $q$. If $q=1$
then the distribution is transformed into Maxwellian distribution. Also, $q<1$
and $q>1$ represent super-extensive and sub-extensive distributions,
respectively. We considered super extensive distribution for all the species
and for convenience we have consider the same values of $q-$ parameters in the
section 5. Dust charging frequency is $10^{4}-10^{5}$ times higher than the
frequency of dust acoustic waves(DAWs) in this type of plasma. Therefore, in
our derivation, we have ignored the effect of dust charge fluctuations.

One more thing worth mentioning here is the Debye radius of dusty
plasma($\lambda_{D}$). As we consider the model as an electron-positron-ion-
dust (EPID) model then the $\lambda_{D}$ can be expressed as below [1],

$\frac{1}{{\lambda_{D}}^{2}}=\frac{1}{{\lambda_{De}}^{2}}+\frac{1}{{\lambda_{Dp}}^{2}}+\frac{1}{{\lambda_{Di}}^{2}}$
(8)

where $\lambda_{De}=\left(\frac{T_{e}}{4\pi n_{e0}e^{2}}\right)^{1/2}$,
$\lambda_{Dp}=\left(\frac{T_{p}}{4\pi n_{p0}e^{2}}\right)^{1/2}$ and
$\lambda_{Di}=\left(\frac{T_{i}}{4\pi n_{i0}e^{2}}\right)^{1/2}$ are the Debye
radius of electrons, protons, and ions, respectively. Here, $T_{e}$, $T_{p}$,
and $T_{i}$ are the temperatures of the species. Also, $n_{e0}$, $n_{p0}$, and
$n_{i0}$ are the unperturbed electrons, positrons, and ions number density
respectively. Where electrons and positrons are created by pair production
there all the parameters except the charge are the same for these two species.
So if the temperatures and densities of electrons and ions are almost equal
then the Debye radii are also equivalent to each other. But we are working in
a region where $Tp=Te>>T_{i}$ and ion density is greater than the density of
the electrons and positrons. So in our case
$\lambda_{D}\sim\lambda_{Di}>>\lambda_{De}$.

##  3 Linear and Nonlinear Analysis:

For the linear and nonlinear analysis of the dust acoustic mode in four
component electron-positron-ion-dust plasma we consider the reductive
perturbation of the field quantities $n_{d}$, $u_{d}$, and $\varphi$.

$n_{d}=1+\sum_{m=1}^{\infty}\epsilon^{m}{n_{d}}^{(m)}$ (9)
$u_{d}=u_{0}+\sum_{m=1}^{\infty}\epsilon^{m}{u_{d}}^{(m)}$ (10)

and,

$\varphi=\sum_{m=1}^{\infty}\epsilon^{m}{\varphi}^{(m)}$ (11)

###  3.1 Linear Dispersion Characteristics:

In order to derive the linear dispersion characteristics we assume all the
field variables are varying as $expi(kx-\omega t)$, now using the equations
(9-11) we get,

$a\omega^{3}+b\omega^{2}+c\omega+d=0$ (12)

The constants $a,b,c$ and $d$ are given by,

$\left.\begin{array}[]{ll}\displaystyle
a=\left(k^{2}\tau_{m}+\tau_{m}D\right)\\\\[8.0pt] \displaystyle
b=\left(k^{2}+D\right)\left[\tau_{m}ku_{0}-1\right]\\\\[8.0pt] \displaystyle
c=\left(k^{2}+D\right)\left[2ku_{0}-\tau_{m}u_{0}k^{2}-2\tau_{m}k^{2}{u_{0}}^{2}-k^{2}\eta_{d}\right]+\tau_{m}k^{2}\\\\[8.0pt]
\displaystyle
d=\left(k^{2}+D\right)\left[k^{3}u_{0}\eta_{d}+\tau_{m}{u_{0}}^{2}k^{3}-k^{2}{u_{0}}^{2}\right]+k^{2}-\tau_{m}u_{0}k^{3}\\\\[8.0pt]
\end{array}\right\\}$ (13)

where, $D$ in the equation 13 is

$D=\frac{{\alpha_{e}}^{2}\sigma_{e}}{2}(1+q_{e})+\frac{{\alpha_{i}}^{2}\sigma_{i}}{2}(1+q_{i})+\frac{{\alpha_{p}}^{2}\sigma_{p}}{2}(1+q_{p})$
(14)

Clearly, the dispersion relation has three roots, by solving the equation (12)
we get,

$\left.\begin{array}[]{ll}\displaystyle\omega_{1}=S+T-\frac{b}{3a}\\\\[8.0pt]
\displaystyle\omega_{2}=-\frac{S+T}{2}-\frac{b}{3a}+\frac{i\sqrt{3}}{2}(S-T)\\\\[8.0pt]
\displaystyle\omega_{3}=-\frac{S+T}{2}-\frac{b}{3a}-\frac{i\sqrt{3}}{2}(S-T)\\\\[8.0pt]
\end{array}\right\\}$ (15)

A full solution with the values of $S$ and $T$ is given in the A.

###  3.2 KdVB and KdV Equations:

In order to investigate the nonlinear behavior of dust acoustic waves the
reductive perturbation techniques of the field quantities given in the
equations (9-11) have been used. Also, we use the stretching of space, time
variables, and viscous, coupling parameters as follows,

$\left.\begin{array}[]{ll}\displaystyle\xi=\epsilon^{1/2}\left(x-Mt\right)\\\\[8.0pt]
\displaystyle\tau=\epsilon^{3/2}t\\\\[8.0pt]
\displaystyle\tau_{m}=\epsilon^{1/2}\tau_{0}\\\\[8.0pt]
\displaystyle\eta_{d}=\epsilon^{1/2}\eta_{0}\\\\[8.0pt] \end{array}\right\\}$
(16)

Where, $M$ is the Mach number and $\epsilon$ is the strength of the
nonlinearity $(0<\epsilon<1)$.

We yield the KdV-Burgers’ equation as follows,

$\frac{\partial\varphi^{(1)}}{\partial\tau}+A\varphi^{(1)}\frac{\partial\varphi^{(1)}}{\partial\xi}+B\frac{\partial^{3}\varphi^{(1)}}{\partial\xi^{3}}-C\frac{\partial^{2}\varphi^{(1)}}{\partial\xi^{2}}=0$
(17)

Where, $A$, $B$, and $C$ are the nonlinear, dispersive, and dissipative
coefficients, respectively. They are given below

$\left.\begin{array}[]{ll}\displaystyle
A=B\Bigg{[}\sum_{j=i,p,e}(\pm)_{j}\alpha_{j}\sigma_{j}^{2}\frac{(1-q_{j})(3-q_{j})}{8}-\frac{3}{\left(M-u_{0}\right)^{4}}\Bigg{]}\\\\[8.0pt]
\displaystyle B=\frac{(M-u_{0})^{3}}{2}\\\\[8.0pt] \displaystyle
C=\frac{\eta_{0}}{2}\\\\[8.0pt] \end{array}\right\\}$ (18)

The sign of $A$ will be $+$ for $j=i$ and $p$ and $-$ for $j=e$.

This KdVB equation (17) has no known exact solution but there are some special
solutions to this equation. The dissipative coefficient is only dependent on
the viscous coefficient which means if the viscosity of the medium increases,
the dissipative nature of the system also increases. The dispersive
coefficient is depending on the Mach number and the streaming velocity of the
dust particles. The nonlinear term depends on $M$, $u_{0}$, $\alpha_{j}$,
$\sigma_{j}$ , and $q_{j}$ $(j=i,p$ and $e$).

Following the procedure given by Pakzad and Zavidan [43] we derived two
solutions to the equation:

a. For $C^{2}<<4BM$ we get the oscillatory shock wave (OSW) solution given by,

$\varphi^{(1)}=\varphi_{c}+\varphi_{0}exp\left(\frac{C}{2B}\psi\right)cos\sqrt{\frac{M}{B}}\psi$
(19)

b. For $C^{2}>4BM$ we get the monotonic shock wave (MSW) solution given by,

$\varphi^{(1)}=\frac{12A}{B}\left[1-tanh^{2}\psi\right]-\frac{36C}{15A}tanh\psi$
(20)

Clearly, If there is no dissipation that is no viscous effect $(\eta_{0}=0)$
in the present case, then the KdVB equation (17) is transformed into the KdV
equation,

$\frac{\partial\varphi^{(1)}}{\partial\tau}+A\varphi^{(1)}\frac{\partial\varphi^{(1)}}{\partial\xi}+B\frac{\partial^{3}\varphi^{(1)}}{\partial\xi^{3}}=0$
(21)

The solution of this KdV equation would give us the solitary structures given
by,

$\varphi^{(1)}=\frac{3M}{A}sech^{2}\left(\frac{\psi}{2\sqrt{\frac{M}{B}}}\right)$
(22)

##  4 Coupling in the liquid-crystal region:

###  4.1 Reason of considering another process:

The effect of coupling is very important for the description of the
crystallization of a dusty plasma system. In fact, it is the main reason for
forming this type of ordered structure in a dusty plasma medium. But, from the
equations (17) and (18) it can be clearly observed that there is no visible
effect of the viscoelastic relaxation time in the nonlinear KdVB equation. So
there may be two possibilities,

a. Coupling does not have any effect on the nonlinear structures.

b. The effect of coupling is absorbed by the other variables and coefficients.

If we consider the first reason is true then the model given above will be
valid in a very limited region. But various authors [44] have shown that
models like the above mentioned are valid at least up to
$\Gamma\sim\Gamma_{c}$. So coupling must have some effect on the nonlinear
structures too. From the work of previous scientists [44] we can write the
expression of the viscoelastic relaxation time as

$\tau_{m}=\eta_{d}\frac{T_{e}}{T_{d}}\left[-0.396\Gamma+0.739\Gamma^{\frac{1}{4}}+0.058\Gamma^{-\frac{1}{4}}-0.27\right]^{-1}$

From this equation, it is very clear that the viscous coefficient has a direct
relation with $\tau_{m}$ so the second reason is the correct one. Also as we
have considered the stretching of the space and time-like variables along with
the viscoelastic relaxation time that is why the effect of the coupling
coefficients may have been absorbed by the other variables. So to find out the
effect of the coupling individually of the system with higher $\Gamma$ i.e.,
near the liquid crystal region, we have to treat the coupling in a different
way[38].

###  4.2 Liquid-crystal Model:

For the reason mentioned in the section 4.1 we only have to change the
momentum equation of the governing equation, and the new equation is given by,

$n_{d}\Bigg{(}\frac{\partial u_{d}}{\partial t}+u_{d}\frac{\partial
u_{d}}{\partial x}\Bigg{)}=n_{d}\frac{\partial\varphi}{\partial
x}-\frac{\partial}{\partial
x}\left(n_{d}d\right)+\eta_{d}\frac{\partial^{2}u_{d}}{\partial x^{2}}$ (23)

Here, the second term in the right-hand side of the equation (23) is the
representation of strong coupling. Before starting any operation with the help
of this equation we have to understand the factor ‘$d$’. This $d$ can be
expressed as,

$d=d_{0}+\epsilon d_{1}+\epsilon^{2}d_{2}+...$ (24)

where,

$\left.\begin{array}[]{ll}\displaystyle
d_{1}=d_{11}n^{(1)}+d_{12}\varphi^{(1)}\\\\[8.0pt] \displaystyle
d_{2}=d_{21}n^{(2)}+d_{22}\varphi^{(2)}+d_{23}{n^{(1)}}^{2}+d_{24}n^{(1)}\varphi^{(1)}+d_{25}{\varphi^{(1)}}^{2}\\\\[8.0pt]
\end{array}\right\\}$ (25)

Here,

$\left.\begin{array}[]{ll}\displaystyle
d_{11}=d_{21}=\frac{d_{0}}{3}\frac{1+\kappa_{0}+{\kappa_{0}}^{2}}{1+\kappa_{0}}\\\\[8.0pt]
\displaystyle
d_{12}=d_{22}=-d_{0}c_{2}\frac{{\kappa_{0}}^{2}}{1+\kappa_{0}}\\\\[8.0pt]
\displaystyle
d_{23}=\frac{d_{0}}{18}\frac{{\kappa_{0}}^{3}-3{\kappa_{0}}^{2}-2\kappa_{0}-2}{1+\kappa_{0}}\\\\[8.0pt]
\displaystyle
d_{24}=-\frac{d_{0}}{3}c_{2}\frac{{\kappa_{0}}^{2}\left(\kappa_{0}-1\right)}{1+\kappa_{0}}\\\\[8.0pt]
\displaystyle
d_{25}=-\frac{d_{0}}{2}\left(3c_{3}-{c_{2}}^{2}\kappa_{0}\right)\frac{{\kappa_{0}}^{2}}{1+\kappa_{0}}\\\\[8.0pt]
\end{array}\right\\}$ (26)

The values of $c_{2}$ and $c_{3}$ are given in the B. And, the $d$ is the
normalized electrostatic temperature and $\kappa_{0}$ is a dimensionless
quantity which is the ratio of inter-particle distance and the Debye length.
This $\kappa_{0}$ is also the measurement of the coupling strength in the
plasma.

As we mentioned above a model of this type was first developed to define the
crystalline plasma structures but nowadays such models have a wide range of
uses in dusty plasma. We can also say that this is the most general model for
a dusty plasma and will be shown in our discussion in the section 5.
Immediately one question may arise why do we use the previous model? In
section 5 we will answer this question too.

At first, we calculated the linear dispersion relation for this type of
mathematical representation of the coupling effect. We get the dispersion
relation as,

$a_{1}\omega^{2}+b_{1}\omega+c_{1}=0$ (27)

Here,

$\left.\begin{array}[]{ll}\displaystyle a_{1}=\left(k^{2}+D\right)\\\\[8.0pt]
\displaystyle b_{1}=-\left(2ku_{0}+k^{2}\eta_{0}\right)a_{1}\\\\[8.0pt]
\displaystyle
c_{1}=a_{1}\left(k^{2}{u_{0}}^{2}+k^{3}\eta_{0}u_{0}-k^{2}\beta\right)-\alpha
k^{2}\\\\[8.0pt] \end{array}\right\\}$ (28)

where, $D$ is previously given in equation (14) and $\alpha$ & $\beta$ is
given below,

$\left.\begin{array}[]{ll}\displaystyle\alpha=1-d_{12}\\\\[8.0pt]
\displaystyle\beta=d_{0}+d_{11}\\\\[8.0pt] \end{array}\right\\}$ (29)

Solving the equation 27 we get,

$\left.\begin{array}[]{ll}\displaystyle{\omega_{1}}^{\prime}=\frac{-b_{1}+\sqrt{{b_{1}}^{2}-4a_{1}c_{1}}}{2a_{1}}\\\\[8.0pt]
\displaystyle{\omega_{2}}^{\prime}=\frac{-b_{1}-\sqrt{{b_{1}}^{2}-4a_{1}c_{1}}}{2a_{1}}\\\\[8.0pt]
\end{array}\right\\}$ (30)

The most important thing from this calculation is the effect of coupling in
the nonlinear regime. So with the help of the previously used reductive
perturbation technique with the newly used equation (24) we get another KdVB
equation,

$\frac{\partial\varphi^{(1)}}{\partial\tau}+A_{1}\varphi^{(1)}\frac{\partial\varphi^{(1)}}{\partial\xi}+B_{1}\frac{\partial^{3}\varphi^{(1}}{\partial\xi^{3}}-C_{1}\frac{\partial^{2}\varphi^{(1)}}{\partial\xi^{2}}=0$
(31)

Here,

$\left.\begin{array}[]{ll}\displaystyle
A_{1}=\frac{-\Lambda\left(\alpha-2d_{12}\alpha-2d_{24}\alpha-2d_{25}\Lambda\right)-2\left(d_{11}+d_{23}\right)\alpha^{2}+2\alpha^{2}\left(M-u_{0}\right)^{2}\Lambda+2c_{2}\Lambda^{3}}{2\alpha\left(M-u_{0}\right)\Lambda}\\\\[8.0pt]
\displaystyle B_{1}=\frac{\Lambda^{2}}{2\alpha\left(M-u_{0}\right)}\\\\[8.0pt]
\displaystyle C_{1}=\frac{\eta_{0}}{2\Lambda}\\\\[8.0pt] \end{array}\right\\}$
(32)

where, $\Lambda=\left[\beta-\left(M-u_{0}\right)^{2}\right]$ and other
quantities are given in the equations (26) and (29). The solution of equation
(31) will be the same as given in the equations (19) and (20).

##  5 Result:

In this paper, we mainly take physically acceptable values of plasma
parameters for astrophysical plasma systems. The values of $M$, $u_{0}$,
$\eta_{0}$, $q$ and $\kappa_{0}$ have given in the table 1.

Plasma Parameters | Range(Normalized) | Plasma | References
---|---|---|---
$M$ |  $1.2$-$3.5$ | Saturn’s Orbit | [45]
$u_{0}$ |  $0.01$-$0.6$ | Laboratory | [46, 47]
$\eta_{0}$ |  $0.15$-$0.55$ & $3$-$5$ | Theoretical & Astrophysical | [37, 41, 43]
$q$ |  $0.459$-$0.846$ | Saturnian MSp $e^{-}$, $H^{+}$ & $O^{+}$ | [48, 49]
$\kappa_{0}$ |  $0.97$, $3.22$, $5.80$ | White dwarf, Jovian interior | [50, 51, 52, 53]
Table 1: Important Plasma Parameters for various Dusty Plasma Systems

In this section, we will mainly discuss the effects of two types of treatments
of the coupling effect. With the help of the calculations in the previous
sections and from the figures here, we will try to answer all the questions we
have raised in the introduction section [section 1].

Before going to that part we will first discuss two things

a) In which equation did the main difference between the two techniques
incorporate?

b) Why have the coupling of electrons and ions not been taken?

To answer the first question we can say that the main difference between these
two treatments is in the momentum equations given in equations (3) and (23).
In both cases, we have considered the electrons and ions to be weakly coupled
as their temperatures are very high compared to the dust grains.

Now we will try to discuss the answer to the first question of the
introduction. The model with the equation (3) was developed to analyze the
plasma system where plasma is treated as a fluid. The second treatment of the
coupling was initially developed for crystalline plasma structure but later it
was also used in the fluid section [ equation 23]. In the next two subsections
5.1 and 5.2 we will discuss the linear and nonlinear waves respectively for
the two types of treatments.

###  5.1 Discussions on Linear waves:

Here, at first, we will discuss the advantage of equation (3) over equation
(23). The dispersion relation arises with the equation (3) is (12) and the
dispersion relation arises with the help of the equation (23) is equation
(27). Clearly, the equation (12) has three roots which are shown in equation
(15) among them two roots $\omega_{1}$ and $\omega_{2}$ are the stable roots.
Figure (1(a)) represents these two stables roots for strong coupling region.
From these two roots, $\omega_{1}$ can be stated as the “slow” mode whereas
$\omega_{2}$ can be described as the “fast” mode of the dust acoustic waves.
Now. if we give attention to figures (1(b)),(1(c)),(2(c)), and (3(b)) we can
see that there is only one mode is there. Because between the two roots of the
equation 27 only one root ${\omega_{1}}^{\prime}$ is the stable one.

Now, if we compare the three figures of (1), where figure (1(a)) arises from
equation (15) and the rest of the two figures (1(b)) and (1(c)) emerge from
equation 27, we can see that the slow mode of figure (1(a)) has the exact same
nature as that of the rest two figures. From this, we can clearly say that the
stable root of equation (27) represents only the “slow” mode. Again only one
curve, i.e., “without coupling” curve of the figure (1(b)) did not have any
negative slope but all the other curves of figure 1 including “with coupling”
curve of the figure (1(b)) have some region with negative slope. In the figure
(1(c)) the dispersion relation has been shown for different values of the dust
longitudinal viscosity and the dispersion curves become steeper with the
increase of the value of viscosity. So the probability of viscous damping
becomes greater than the Landau-Vlasov damping [54]. On the other hand, the
effect of the viscous coefficients on the dispersion relation (12) is
negligible as the value of $\tau_{0}$ mainly depends on the dust temperature
$T_{d}$. The dust temperatures have a very low value in comparison to that of
the other species and the value of $\tau_{0}$ is inversely proportional to the
dust temperature.

(a)

(b)

(c)

Figure 1: (a) Two roots of the dispersion relation (12) (b) Dispersion
relation plot from equation (27) with and without coupling (c) Dispersion
relation plot from equation (27) with different values of the dust
longitudinal viscosity.

In figure 2 we have shown the dispersion relation of two stable modes of (12)
and one stable mode of (27) due to different streaming velocities of dust. The
slow mode of equation (12) (figure 2(b)) and only stable mode of equation (27)
(figure 2(c)) have the same nature whereas the fast mode of equation (12) has
the opposite nature (figure 2(a)). By a careful look at the figures (2(a)) and
(2(b)) we can see that the fast and slow modes intersect each other for the
streaming velocity $u_{0}=0.09$.

(a)

(b)

(c)

Figure 2: Dispersion relation for different streaming velocities of dust

In figure (3(a)) we have shown the change in the dispersion relations due to
coupling in the first case. In the strongly coupled region, the fast mode of
the equation (12) has the negative slope $(\frac{d\omega(k)}{dk}<0)$. In
figure (3(b)) we can see the slopes of the curves positive for most of the
time and the amount of slope is increasing with the increase of the lattice
parameter $(\kappa_{0})$.

(a)

(b)

Figure 3: (a) Dispersion relation of the first kind for weak and strong
coupling (b) Dispersion relation for second for different lattice parameter
$\kappa_{0}$ at $T_{*0}$

###  5.2 Discussions on Non-linear waves:

Now, we will discuss the nonlinear waves for both the treatments shown in
sections 3.2 and 4. From equation (16) we can see that viscoelastic relaxation
time $(\tau_{m})$ is used as a stretched quantity whereas from equation (24)
we can see that the normalized electrostatic temperature $(d)$ is treated as
perturbative quantity. Both of these two considerations lead to the KdVB
equation but with a very important difference. That important difference is
one of the main focus of this work. If we look at the equation (18) which
represents the coefficients of the KdVB equation for the first kind of
treatment the effect of coupling is absent as coefficients are independent of
$\tau_{0}$. But if we look at the equation (32) we can see that the
coefficients have a direct dependence on coupling in terms of electrostatic
temperature. The meaning of direct dependence is that in the first kind of
treatment, though there is no coefficient containing viscoelastic relaxation
time, there must be a dependency on coupling as there is a term of viscosity
in the KdVB equation. Now we will discuss the variation of the shock waves
with very important plasma parameters in this context. In figure (4) we have
described the effect of the longitudinal viscous coefficient on two types of
shock waves. When the viscous coefficient has a small value we get the
oscillatory shock waves (OSW) for both the treatments. It is quite obvious
that the increase of the viscous coefficient is increasing the effect of the
shock for OSW and it is shown for both the figures (4(a)) and (4(b)). Another
important factor we want to describe here for the same two figures is (4(a))
and (4(b)). The wavelength of the figure (4(a)) is greater than the wavelength
of the figure (4(b)). Also, the amplitude of the first one is a little bit
less than the second one. These two effects have an explanation that the
effect of coupling for the first kind is absorbed by the rest of the
parameters but the same effect arises in the second case independently. The
difference due to different treatments on coupling coefficients has also been
observed for the monotonic shock waves (MSWs) ( figures 4(c) and 4(d)).

(a)

(b)

(c)

(d)

Figure 4: Oscillatory Shock Waves for (a) first kind (b) second kind and
Monotonic Shock Waves for (c) first kind (d) second kind with different
viscous coefficients

Also, the normalized electrostatic temperature has relations with the lattice
parameter $(\kappa_{0})$ (Equation 26). In figure (5) we have shown the
variation of OSWs and MSWs for different $d_{0}$ and $\kappa_{0}$. With the
increase of both the parameters, the amplitude and the wavelength of the OSW
are increasing. The MSWs are also increasing the strength of the shock for
both parameters. Physically we can say that the increase of the electrostatic
temperature and lattice parameters when all the other parameters are constant
leads to a decrease in the coupling of the system and that is why the dust
particles can move more freely. It is worth mentioning that the value
$\kappa_{0}=5.80$ has a greater effect on both types of shocks.

(a)

(b)

(c)

(d)

Figure 5: Oscillatory Shock Waves for different normalized (a) electrostatic
temperature $(d_{0})$ (b) lattice parameter $(\kappa_{0})$ and Monotonic Shock
Waves for different normalized (c) electrostatic temperature $(d_{0})$ (d)
lattice parameter $(\kappa_{0})$

(a)

(b)

Figure 6: Oscillatory Shock Waves for different Mach number in (a) first kind
(b) second kind

(a)

(b)

Figure 7: Monotonic Shock Waves for different Mach number in (a) first kind
(b) second kind

(a)

(b)

Figure 8: Oscillatory Shock Waves for different Streaming Velocity in (a)
first kind (b) second kind

(a)

(b)

Figure 9: Monotonic Shock Waves for different Mach number in (a) first kind
(b) second kind

Among all the common plasma parameters of the two treatments the Mach number
$(M)$ and the streaming velocity $(u_{0})$ are another two very important
parameters. From figure (6) to figure (9) we have discussed their importance.
Let us first discuss the OSWs for different values of Mach numbers in both
treatments. From figures (6(a)) and (6(b)) it is shown that the amplitude and
wavelengths both increase for the first kind whereas the same parameters
decrease for the second case. Now, if we give attention to the figures (7(a))
and (7(b)) we can observe that the amount of shock increases for the first
case and decreases for the second case. Also in the figure (7(b)) there is a
special phenomenon that can be seen that the curves for $M=1.3$ and $M=1.4$
cut the curve $M=1.2$ at least two times. Again figures (6(a)) and (7(a)) are
analogous to the result shown in the figures (4(a)) and (4(d)). The same
effects of the Mach number and viscous coefficient can be seen in all types of
weakly coupled plasmas [54, 55, 41] but Mach number and the viscous
coefficient have opposite responses on the shock if plasma is coupled anyway,
e.g. via magnetic field [40].

If we observe the OSWs (figures 8(a) and 8(b)) and MSWs (figures 9(a) and
9(b))for different streaming velocities in both the treatments we can see that
they have the opposite effects than the effects due to different Mach numbers.
Also, OSWs and MSWs for different dust streaming velocities of the first kind
have opposite effects than the effects of viscous coefficients and this is
again analogous with the effects for weakly coupled plasmas. So the most
important statement we are making in this manuscript for the first time is in
the form of two conditions which universally effective for all plasma systems.

#### Condition 1:

When we treat the strong coupling of a plasma system in the form of a
viscoelastic relaxation time then all the other plasma parameters have shown
their effect as if they are operating in weakly coupled plasma. Because the
effect of the viscoelastic relaxation time has been absorbed by the
longitudinal viscous coefficient.

#### Condition 2:

When we treat the strong coupling of a plasma system in the form of a
normalized electrostatic temperature then all the other parameters have shown
their effect as if they are operating in a strongly coupled plasma. As the
coupling effect is independently present in the system.

_Condition 1_ can only be different if any force (mainly dissipative force
like ‘collision’) is coupled with viscoelastic relaxation time itself.

Now, we will discuss the effect of the rest of the plasma parameters on the
OSW and MSW. As we consider the temperature of dust is $100$ and $1000$ times
lesser than the temperature of ions and electrons, respectively, the effect of
the different $q-$ values and densities for both treatments has a minimal
effect. We have shown these effects in the supplementary file, where we can
see the effect of the $q$ values of electrons have a different effect on the
OSWs and MSWs than the $q$ values of positrons and ions.

##  6 Conclusion:

In this paper, we have investigated the dust acoustic shock and solitary waves
associated DAWs in an EPIDPM. To the best of our knowledge, no previous worker
has ever tried two treatments for the coupling for the same plasma system.
From the results, we can see that there are three most important plasma
parameters common for both the treatments, these are Mach number $(M)$, the
streaming velocity of dust $u_{0}$, and the longitudinal viscous coefficient
$\eta_{0}$. Depending on the fact that the coupling term is directly
associated with the longitudinal viscous coefficient or not determines the
behavior of the rest of the parameters. So the first treatment in the
nonlinear region is only effective when the density is very low of the plasma
system and longitudinal viscous force is the only dissipative force present in
the system. To encounter a high-density plasma by this treatment we have to
consider another dissipative force like collision. However, the nonlinear
behavior in the liquid crystal region can not be analyzed even after
considering the collision or collision-like dissipative force. But from the
discussion on the second treatment of the coupling, we can see the nonlinear
effect in all the regions from liquid-crystal to the hydrodynamic phase of a
dusty plasma system.

But there is an advantage of proceeding through the first treatment that is
the linear behavior of dust acoustic waves can be better analysed by this. So
the linear instabilities, e.g. two-stream instability can be understood more
rigorously by the first treatment.

There is one very important finding we have from the supplementary file that
is given below as a note.

_Note:_ the nature of the $q-$ values of ions and positrons are the same and
it is different from the $q-$ value of electrons.

The region of our focus is the magnetosphere region of Saturn. However, we
have not considered the magnetic field in the calculation, which is our future
perspective. The study of different types of instabilities is another path
that we can explore for strongly coupled plasma.

## References:

## References

  * [1] PK Shukla.  A survey of dusty plasma physics.  Physics of Plasmas, 8(5):1791–1803, 2001.
  * [2] C. K. Goertz.  Dusty plasmas in the solar system.  Reviews of Geophysics, 27(2):271–292, 1989.
  * [3] Paval Bliokh, Victor Sinitsin, Victoria Yaroshenko, and D. A. Mendis.  Dusty and self‐gravitational plasmas in space.  Physics Today, 50(2):68–68, 1997.
  * [4] P K Shukla, D A Mendis, and V W Chow.  The Physics of Dusty Plasmas.  WORLD SCIENTIFIC, 1996.
  * [5] D.A. Mendis and M. Rosenberg.  Some aspects of dust-plasma interactions in the cosmic environment.  IEEE Transactions on Plasma Science, 20(6):929–934, 1992.
  * [6] D. A. Mendis and M. Rosenberg.  Cosmic dusty plasma.  Annual Review of Astronomy and Astrophysics, 32(1):419–463, 1994\.
  * [7] P. K. Shukla, D. A. Mendis, and T. Desai.  Advances in Dusty Plasmas, pages 3–19.  WORLD SCIENTIFIC, 1997.
  * [8] John Y. N. Cho and Michael C. Kelley.  Polar mesosphere summer radar echoes: Observations and current theories.  Reviews of Geophysics, 31(3):243–265, 1993.
  * [9] Qihou H. Zhou and Michael C. Kelley.  Meteor observations by the arecibo 430 mhz incoherent scatter radar. ii. results from time-resolved observations.  Journal of Atmospheric and Solar-Terrestrial Physics, 59(7):739–752, 1997.
  * [10] H. M. Mott-Smith and Irving Langmuir.  The theory of collectors in gaseous discharges.  Phys. Rev., 28:727–763, Oct 1926.
  * [11] MS Sodha and S Guha.  Advances in plasma physics.  Wiley, New York, 4:219, 1971.
  * [12] M. Rosenberg and D.A. Mendis.  Uv-induced coulomb crystallization in a dusty gas.  IEEE Transactions on Plasma Science, 23(2):177–179, 1995.
  * [13] M. Rosenberg, D.A. Mendis, and D.P. Sheehan.  Positively charged dust crystals induced by radiative heating.  IEEE Transactions on Plasma Science, 27(1):239–242, 1999.
  * [14] E. C. Whipple, T. G. Northrop, and D. A. Mendis.  The electrostatics of a dusty plasma.  Journal of Geophysical Research: Space Physics, 90(A8):7405–7413, 1985.
  * [15] R Paul, G Sharma, K Deka, S Adhikari, R Moulick, S S Kausik, and B K Saikia.  Experimental study of charging of dust grains in the presence of energetic electrons.  Plasma Physics and Controlled Fusion, 64(3):035009, jan 2022.
  * [16] JR Asbridge, SJ Bame, and IB Strong.  Outward flow of protons from the earth’s bow shock.  Journal of Geophysical Research, 73(17):5777–5782, 1968.
  * [17] Yoshifumi Futaana, Shinobu Machida, Yoshifumi Saito, Ayako Matsuoka, and Hajime Hayakawa.  Moon-related nonthermal ions observed by nozomi: Species, sources, and generation mechanisms.  Journal of Geophysical Research: Space Physics, 108(A1):SMP–15, 2003.
  * [18] M Horányi, G Morfill, and E Grün.  Mechanism for the acceleration and ejection of dust grains from jupiter’s magnetosphere.  Nature, 363(6425):144–146, 1993.
  * [19] SM Krimigis, JF Carbary, EP Keath, TP Armstrong, LJ Lanzerotti, and G Gloeckler.  General characteristics of hot plasma and energetic particles in the saturnian magnetosphere: Results from the voyager spacecraft.  Journal of Geophysical Research: Space Physics, 88(A11):8871–8892, 1983.
  * [20] M Horányi, TW Hartquist, O Havnes, DA Mendis, and GE Morfill.  Dusty plasma effects in saturn’s magnetosphere.  Reviews of Geophysics, 42(4), 2004.
  * [21] Steen H Hansen.  Cluster temperatures and non-extensive thermo-statistics.  New Astronomy, 10(5):371–377, 2005.
  * [22] Constantino Tsallis.  Possible generalization of boltzmann-gibbs statistics.  Journal of statistical physics, 52:479–487, 1988.
  * [23] Jyotirmoy Goswami, Swarniv Chandra, and B Ghosh.  Study of small amplitude ion-acoustic solitary wave structures and amplitude modulation in e–p–i plasma with streaming ions.  Laser and Particle Beams, 36(1):136–143, 2018.
  * [24] W Pilipp.  3.3. 5.2 interplanetary plasma and magnetic field (solar wind) 3.3. 5 interplanetary particles and magnetic field.  Methods, Constants, Solar System, pages 245–251, 1981.
  * [25] Mihály Horányi.  Charged dust dynamics in the solar system.  Annual review of astronomy and astrophysics, 34(1):383–418, 1996\.
  * [26] RJ Murphy, GH Share, JG Skibo, and B Kozlovsky.  The physics of positron annihilation in the solar atmosphere.  The Astrophysical Journal Supplement Series, 161(2):495, 2005.
  * [27] WH Zurek.  Annihilation radiation from the galactic center-positrons in dust?  The Astrophysical Journal, 289:603–608, 1985.
  * [28] Tracy A Ellis and John S Neff.  Numerical simulation of the emission and motion of neutral and charged dust from p/halley.  Icarus, 91(2):280–296, 1991.
  * [29] D Asoka Mendis and Marlene Rosenberg.  Cosmic dusty plasma.  Annual Review of Astronomy and Astrophysics, 32(1):419–463, 1994\.
  * [30] V Nosenko, S Zhdanov, AV Ivlev, G Morfill, J Goree, and A Piel.  Heat transport in a two-dimensional complex (dusty) plasma at melting conditions.  Physical review letters, 100(2):025003, 2008.
  * [31] K Avinash, RL Merlino, and PK Shukla.  Anomalous dust temperature in dusty plasma experiments.  Physics Letters A, 375(30-31):2854–2857, 2011.
  * [32] M Rosenberg, DA Mendis, and DP Sheehan.  Positively charged dust crystals induced by radiative heating.  IEEE transactions on plasma science, 27(1):239–242, 1999.
  * [33] Gadadhar Banerjee and Sarit Maitra.  Arbitrary amplitude dust ion acoustic solitons and double layers in the presence of nonthermal positrons and electrons.  Physics of Plasmas, 23(12):123701, 2016.
  * [34] Ashesh Paul and Anup Bandyopadhyay.  Dust ion acoustic solitary structures in presence of nonthermal electrons and isothermal positrons.  Astrophysics and Space Science, 361:1–9, 2016.
  * [35] Md Habibur Rahman, Nure Alam Chowdhury, Abdul Mannan, and AA Mamun.  Dust-acoustic rogue waves in an electron-positron-ion-dust plasma medium.  Galaxies, 9(2):31, 2021.
  * [36] A.A. Mamun, B. Eliasson, and P.K. Shukla.  Dust-acoustic solitary and shock waves in a strongly coupled liquid state dusty plasma with a vortex-like ion distribution.  Physics Letters A, 332(5):412–416, 2004.
  * [37] EI El-Awady and M Djebli.  Dust-acoustic waves in strongly coupled dusty plasmas with nonextensive electrons and ions.  Canadian Journal of Physics, 90(7):675–681, 2012.
  * [38] S. E. Cousens, S. Sultana, I. Kourakis, V. V. Yaroshenko, F. Verheest, and M. A. Hellberg.  Nonlinear dust-acoustic solitary waves in strongly coupled dusty plasmas.  Phys. Rev. E, 86:066404, Dec 2012.
  * [39] E. S. Belenkaya, I. I. Alexeev, V. V. Kalegaev, and M. S. Blokhina.  Definition of saturn’s magnetospheric model parameters for the pioneer 11 flyby.  Annales Geophysicae, 24(3):1145–1156, 2006.
  * [40] Jyotirmoy Goswami, Swarniv Chandra, Jit Sarkar, and Basudev Ghosh.  Electron acoustic solitary structures and shocks in dense inner magnetosphere finite temperature plasma.  Radiation Effects and Defects in Solids, 175(9-10):961–973, 2020\.
  * [41] J Goswami and S S Kausik.  Investigation of dust ion acoustic shock and solitary waves in a viscous dusty plasma.  Physica Scripta, 98(3):035602, feb 2023.
  * [42] A. H. Sulaiman, A. Masters, and M. K. Dougherty.  Characterization of saturn’s bow shock: Magnetic field observations of quasi-perpendicular shocks.  Journal of Geophysical Research: Space Physics, 121(5):4425–4434, 2016.
  * [43] Hamid Reza Pakzad and Kurosh Javidan.  Dust acoustic solitary and shock waves in strongly coupled dusty plasmas with nonthermal ions.  Pramana, 73(5):913–926, 2009.
  * [44] P. K. Kaw.  Collective modes in a strongly coupled dusty plasma.  Physics of Plasmas, 8(5):1870–1878, 05 2001.
  * [45] E. Echer.  Solar wind and interplanetary shock parameters near saturn’s orbit.  Planetary and Space Science, 165:210–220, 2019.
  * [46] Robert L. Merlino.  Dust-acoustic waves driven by an ion-dust streaming instability in laboratory discharge dusty plasma experiments.  Physics of Plasmas, 16(12):124501, 12 2009.
  * [47] Yoshifumi Saitou and Osamu Ishihara.  Dynamic behavior of dust particles in plasmas.  In Tetsu Mieno, Yasuaki Hayashi, and Kun Xue, editors, Progress in Fine Particle Plasmas, chapter 4. IntechOpen, Rijeka, 2020.
  * [48] K Dialynas, SM Krimigis, DG Mitchell, DC Hamilton, N Krupp, and PC Brandt.  Energetic ion spectral characteristics in the saturnian magnetosphere using cassini/mimi measurements.  Journal of Geophysical Research: Space Physics, 114(A1), 2009.
  * [49] P Schippers, M Blanc, N André, I Dandouras, GR Lewis, LK Gilbert, AM Persoon, N Krupp, DA Gurnett, AJ Coates, et al.  Multi-instrument analysis of electron populations in saturn’s magnetosphere.  Journal of Geophysical Research: Space Physics, 113(A7), 2008.
  * [50] E. Schatzman.  White dwarfs.  (North-Holland, Amsterdam, 1958.
  * [51] EE Salpeter.  On convection and gravitational layering in jupiter and in stars of low mass.  Astrophysical Journal, vol. 181, p. L83, 181:L83, 1973.
  * [52] Setsuo Ichimaru.  Strongly coupled plasmas: high-density classical plasmas and degenerate electron liquids.  Rev. Mod. Phys., 54:1017–1059, Oct 1982.
  * [53] O. S. Vaulina and S. A. Khrapak.  Scaling law for the fluid-solid phase transition in yukawa systems (dusty plasmas).  Journal of Experimental and Theoretical Physics, 90(2):287–289, Feb 2000.
  * [54] Jyotirmoy Goswami, Swarniv Chandra, and Basudev Ghosh.  Shock waves and the formation of solitary structures in electron acoustic wave in inner magnetosphere plasma with relativistically degenerate particles.  Astrophysics and Space Science, 364(4):65, Apr 2019.
  * [55] J. Goswami, S. Chandra, J. Sarkar, S. Chaudhuri, and B. Ghosh.  Collision-less shocks and solitons in dense laser-produced fermi plasma.  Laser and Particle Beams, 38(1):25–38, 2020.

## Appendix A

The dispersion relation is given in equation 12. These we substitute,

$\omega=y-\frac{b}{3a}$ (33)

Therefore,

$a\left(y-\frac{b}{3a}\right)^{3}+b\left(y-\frac{b}{3a}\right)^{2}+c\left(y-\frac{b}{3a}\right)+d=0$
(34)

Rearranging this we get,

$y^{3}+\left(\frac{3ac-b^{2}}{3a^{2}}\right)y+\frac{2b^{3}-9abc+27a^{2}d}{27a^{3}}=0$
(35)

Putting $Q=\frac{3ac-b^{2}}{9a^{2}}$ and
$R=\frac{2b^{3}-9abc+27a^{2}d}{54a^{3}}$ we get,

$y^{3}+3Qy-2R=0$ (36)

Considering the identity
$\left(S+T\right)^{3}-3ST\left(S+T\right)-\left(S^{3}+T^{3}\right)=0$ and
matching with the equation 36 we get,

$\left.\begin{array}[]{ll}\displaystyle y=S+T\\\\[8.0pt] \displaystyle
ST=-Q\\\\[8.0pt] \displaystyle S^{3}+T^{3}=2R\\\\[8.0pt] \end{array}\right\\}$
(37)

Now, proceed with this, and using Vieta’s formula we get the value of $S$ and
$T$ as,

$\left.\begin{array}[]{ll}\displaystyle
S=\sqrt[3]{R+\sqrt{R^{2}+Q^{3}}}\\\\[8.0pt] \displaystyle
T=\sqrt[3]{R-\sqrt{R^{2}+Q^{3}}}\\\\[8.0pt] \end{array}\right\\}$ (38)

Again with some algebraic steps, we get the following solution,

$\left.\begin{array}[]{ll}\displaystyle y_{1}=S+T\\\\[8.0pt] \displaystyle
y_{2}=\frac{-\left(S+T\right)+i\sqrt{3}\left(S-T\right)}{2}\\\\[8.0pt]
\displaystyle
y_{3}=\frac{-\left(S+T\right)-i\sqrt{3}\left(S-T\right)}{2}\\\\[8.0pt]
\end{array}\right\\}$ (39)

Again, from equations 33 and 39 we got the solution given in equation 15.

## Appendix B

The $c_{2}$ and $c_{3}$ in the equation 26 are given below,

$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\Bigg{[}\left(1+q_{e}\right)\left(3-q_{e}\right)\alpha_{e}{\sigma_{e}}^{2}-\left(1+q_{i}\right)\left(3-q_{i}\right)\alpha_{i}{\sigma_{i}}^{2}-\left(1+q_{p}\right)\left(3-q_{p}\right)\alpha_{p}{\sigma_{p}}^{2}\Bigg{]}$

and

$\displaystyle c_{3}$
$\displaystyle=\frac{1}{48}\Bigg{[}\left(1+q_{e}\right)\left(3-q_{e}\right)\left(5-3q_{e}\right)\alpha_{e}{\sigma_{e}}^{3}-\left(1+q_{i}\right)\left(3-q_{i}\right)\left(5-3q_{i}\right)\alpha_{i}{\sigma_{i}}^{3}$
$\displaystyle-\left(1+q_{p}\right)\left(3-q_{p}\right)\left(5-3q_{p}\right)\alpha_{p}{\sigma_{p}}^{3}\Bigg{]}$