Study of Breather Structures in the Framework of Gardner Equation in Electron-
Positron-Ion Plasmas

Snehalata Nasipuri1, Swarniv Chandra2, Uday Narayan Ghosh3∗, Chinmay Das4,
Prasanta Chatterjee1
1\. Department of Mathematics, Visva-Bharati, Santiniketan-731235, India
2\. Department of Physics, Government General Degree College at Kushmandi,
Dakshin Dinajpur-700032, India

3\. Department of Mathematics, K. K. M. College,

A Constituent unit of Munger University, Jamui, Bihar-811307, India
4\. Department of Mathematics, K.J.R. Government General Degree College,
Bankura-722143, India

Abstract

In different nonlinear mediums, the wave trains carry energy and expose many
amazing features. To describe a nonlinear phenomenon, a soliton is one that
preserves its shape and amplitude even after the collision. Breather is one
kind of soliton structure, which is a localized wave that periodically
oscillates in amplitude. In the Korteweg-de Vries (KdV) equation, nonlinearity
appears in quadratic terms, whereas the modified Korteweg-de Vries (mKdV)
equation contains cubic nonlinearity. When we consider both the quadratic and
cubic nonlinearities, we obtain the Gardner equation (GE). So GE is also
called the KdV-mKdV equation. Due to the prominent balance between nonlinear
terms (quadratic and cubic) and higher-order linear terms in an evolution
equation, one can get a soliton profile solution, but when both (quadratic and
cubic) nonlinear terms and higher-order linear terms appear together in an
evolution equation, we are enabled to obtain breather soliton structures. This
article uses the reductive perturbation technique (RPT) to get the GE from a
plasma system with four parts: cold positrons that can move, hot positrons and
hot electrons that are spread out in a kappa pattern, and positive ions that
can’t move. Then, using the Hirota bilinear method (HBM), it is possible to
obtain the multi-soliton and breather structures of GE. Breathers are
fluctuating regional wave packets and significantly participate in
hydrodynamics as well as optics; besides, their interaction can alter the
dynamical characteristics of the wave fields. We also incorporate a detailed
numerical simulation study based on a newly designed code by two of the co-
authors. It is found that in our plasma system, soliton solutions, especially
breather solutions, exist. Although superthermal (kappa-distributed) electrons
and positrons play an important role in soliton structures, This type of
analysis can also apply to the propagation of finite-amplitude waves in
natural phenomena like the atmosphere, ocean, optic fibres, signal processing,
etc. It should also be useful to study different electrostatic disturbances in
space and laboratory plasmas, where immobile positive ions, superthermal
electrons, superthermal hot positrons, and mobile cold positrons are the major
plasma species.

Keywords: Positron acoustic waves; Kappa distribution; Hirota bilinear method;
modified Korteweg-de Vries; Gardner equation; Solitons; Breathers.

*Corresponding author e-mail<EMAIL_ADDRESS>

##  1 Introduction

Nowadays, the study of nonlinear phenomena in electron-positron (e-p) and
electron-positron-ion (e-p-i) plasmas attracts more attention from researchers
due to their existence in astrophysical space plasmas as well as laboratory
plasmas like supernovas, quasar and pulsar magnetospheres, cluster explosions,
active galactic nuclei, Van Allen radiation belts, polar cups of fast rotating
neutron stars, semi-conductor plasmas, and intense laser fields. In linear
theory, the amplitude of the wave is sufficiently small to neglect the
contributions of second- and higher-order terms, i.e., nonlinear terms. In the
case of larger wave amplitudes, the linear approximation breaks down and
nonlinearity is taken into account. The harmonic generation, including fluid
advection, the nonlinear Lorentz force, the trapping of particles in the wave
potential, etc., introduces the nonlinearities. In plasmas, the nonlinear
effect leads to localised waves and creates different types of structures like
soliton solutions, breather solutions, etc. Our main object is to find the
soliton structures and the breather structures for a four-component plasma
system consisting of immobile positive ions. mobile cold positrons and Kappa-
distributed hot positrons and hot electrons, and we are able to establish such
observations by deriving the Gardner equation (GE) from the above-mentioned
model equations. GE is used to model the wave propagation in a one-dimensional
nonlinear lattice with a nonharmonic-bound particle. Since GE is an extension
of KdV and mKdV equations, the properties of GE are obtained by the properties
of the mKdV equation. In this article, we have studied a special nonlinear
feature, which is the breather soliton structure. We know that solitary waves
are localised, travelling steady-profile solutions for dispersive nonlinear
dynamical systems. Breathers are also localised travelling waves, but their
profile is not steady and changes periodically in time; they are a mix of
soliton and periodic solutions. In a coordinate system moving with the wave,
the wave profile is itself a moving periodic travelling wave, containing an
amplitude envelope that keeps it localised in space. It is hard to
characterise breathers analytically, even in (1+1)-dimension, since they are
solutions for which the governing equations can’t be reduced to a lower-order
system. So, to produce breathers in weakly nonlinear limits, where breathers
are of small amplitude, one can look at them with very shallow envelopes; in
this case, the separation of scales allows a reduction of the governing
equations to a lower-order system. So, our study will be helpful for further
investigation of the nonlinear structures (soliton, breathers) for other
evolution equations that can be derived from the mentioned system.
The high-energy particles can be found in space plasmas or laboratory plasmas,
and such high energization may arise due to the effect of external forces
acting on the natural space environment plasmas or wave-particle interaction.
Plamas with superthermal electrons, or positrons, are generally characterised
by a long tail in the high-energy region. A generalised Lorentzian or kappa
distribution is used to model such space plasmas [1]. Many researchers have
studied the nonlinear propagation of waves in e-p-i plasmas [2]-[3]. A study
by Alam et al. [4] looked at the effects of superheated electrons and
positrons on positron-acoustic single waves and double layers in e-p-i
plasmas. A three-dimensional generalised Lorentzian or kappa distribution
function [5] can be written as:

$F_{\kappa}(\nu)=\frac{\Gamma(\kappa+1)}{(\pi\kappa{\theta}^{2})^{(3/2)}\Gamma(\kappa-1/2)}(1+\frac{{\nu}^{2}}{\kappa\theta^{2}})^{-(\kappa+1)},$

where $\Gamma$ is the gamma function, $\theta$ is the most probable speed or
effective thermal speed connected to the usual thermal velocity
$V_{t}=(\frac{k_{B}T}{m})^{(1/2)}$ by $\theta=\frac{2\kappa-3}{\kappa}V_{t},$
$T$ is the characteristic kinetic temperature, and $k_{B}$ is the Boltzmann
constant. The parameter $\kappa$ is the spectral index parameter that defines
the strength of the superthermality. The range of this parameter is
$3/2<\kappa<\infty$. In the limit, $\kappa\to\infty,$ the kappa distribution
function reduces the well-known Maxwell-Boltzmann distribution. So, the kappa
distribution function is a more generalized function than the Maxwellian
distribution function. Kappa distribution has been used to interpret
spacecraft data in the Jupiter and Saturn [6], Earth’s magnetospheric plasma
sheet [7], and the solar wind, to explain the velocity filtration effect in
the solar corona [8], and to analyze the field-aligned conductance values in
the auroral region by using the Freja satellite data [9].
From the initial works on the early universe to present-day laboratory
experiments, electron-positron ion (e-p-i) plasma has been a fundamental field
of interest for theoretical and experimental plasma physicists. E-P-I plasma
has successfully explained where gamma-ray sources come from and how they work
by combining relativistic electrons and positrons in a strong laser field
[10]. Electrons and positrons form an inverted, two-level system. In such
types of studies, Klein-Gordan field theory can be helpful. Later in 2006,
Eliasson and Shukla [11] studied coherent structures in phase-space vortices
in pair plasma. Pitrou et al. [12] have given a picture of the Big Bang
nucleosynthesis, including stages of e-p-i plasma.
Luque and Schamel [13] have theorised that electrostatic trapping can be a key
to the dynamics of plasmas and other similar collective systems. Volker and
Tsupko [14] have carried out a detailed calculation and showed how the e-p-i
plasma influences the black hole shadows. These works also refer to previous
works of Breuer and Ehlers [15]. Ruffini et al. [16] have referred to
electron-positron (e-p) pairs in physics, from heavy nuclei to black holes.
This work refers to gravitational collapse leading to the formation of black
holes observed in gamma-ray bursts (GRBs). Even studies in new fields like the
generation of antihydrogen have electron-positron plasma [17]. Such a study
was instrumental in many stages of research on the antiproton decelerator at
CERN in Geneva. Thus, we see that the study of antimatter also involved
references to the correlative properties of e-p-i plasma ([18],[19]). This
work reviews the recombination mechanisms of electrons and positrons to
produce antihydrogen. The ATHENA and ATRAP collaborations with CERN have dealt
a lot with such e-p pair plasma.
Coming to works of interest we have found how ion acoustic shock waves in
plasmas are formed with warm ions and Kappa-distributed electrons and
positrons (Hussain et al. [20]). Mehdian [21] and collaborators have studied
ion-acoustic waves carrying orbital angular momentum (OAM) in the e-p-i
plasmas. Asit Saha [22] studied the nonlinear excitations for the positron
acoustic shock waves in dissipative nonextensive unmagnetized e-p-i plasmas.
With a magnetized e-p-i plasma two-stream instability has been studied by
Tinakiche et al. [23]. Moslem et al. [24] and others [25] studied nonlinear
excitations in e-p-i plasmas in accretion disks of active galactic nuclei
(AGN). Now, non-Maxwellian electron distribution plays an important role. It
may be Cairsis distributed or super thermally distributed (Kappa, Shamel,
etc). Jilani [26] has shown how electron acoustic waves (EAWs) evolve in
magneto-rotating e-p-i plasma with nonthermal electrons and positrons.
The effect of dust in e-p-i plasmas with superthermal electrons has been also
studied by Siberian et al. [27]. Mugemana et al. [28] have incorporated charge
separation in nonlinear waves in e-p-i plasmas. Furthermore, gyrokinetic
stability analysis in e-p-i plasmas was carried out by Mishchenko et al. [29].
The effect of trapped ions was studied by Alinejad and his team [30]. Two-
dimensional electrostatic shock waves in relativistic e-p-i plasma were
studied by Masood [31]. Heavy ion-acoustic rogue waves in plasmas, which
contain electrons, positrons, and positive and negative ions, were studied by
Chowdhury et al. [32].
Maroof [33] have studied magneto-hydrodynamic (MHD) waves in e-p plasmas with
electron spin. Even non-planar cylindrical fast magneto-sonic solitary waves
in degenerate e-p-i plasma were studied by Abdikian [34]. Recently, Kaur et
al. [35] have studied the ion-acoustic breathers in electron-beam plasma. The
breather structures and peregrine solitons in a polarized space dusty plasma
have been investigated by Saini et al. [36]. Slathia et al. have shown that
dust-acoustic solitons, rogons, and breathers exist in Jupiter’s
magnetospheric dusty plasmas [37].

The motivation behind carrying out this work is the study of field quantities
in an electron-positron-ion plasma over the gradual evolution of stationary
structures. In the past numerous works have been done in lines of regular KdV
equations with various plasma compositions under varied configurations. None
of those works were complete in terms of the field evolutionary mechanism and
its dependence on parameters. As per our knowledge, no work on breather using
HBM considering our proposed model equations has been done in plasma till
today. Since any intermediary stages are not possible to study analytically,
probably the early authors have missed this point. So we have attempted to
study the evolution by numerical means.

The paper is organised in the following way: Section 2, contains the
considered model equations. In section 3, we present the evolutionary
equations starting from the derivation of KdV, modified KdV, and finally the
Gardner equation and provide their analytic solutions in the next section 4.
The basic single soliton solution is presented in subsection 4.1, double
soliton solutions in subsection 4.2, and the breather solution in subsection
4.3. In section 5, we present the analytical (5.1) and numerical (5.2) results
and finally put the summary in the conclusion 6.

##  2 The mathematical model

We consider the non-linear propagation of positron-acoustic waves in a four-
component plasma system consisting of immobile positive ions, mobile cold
positrons and Kappa distributed hot positrons and hot electrons. Hence at
equilibrium $n_{e0}=n_{pc0}+n_{ph0}+n_{i0}$, where $n_{i0}$, $n_{e0}$ are the
unperturbed ion number density and electron number density respectively.
$n_{pc0}(n_{ph0})$ is the number density of unperturbed cold(hot) positron.
The hot electrons and the hot positrons are assumed to obey Kappa distribution
on the positron-acoustic wave time scale and are given by the following
expressions:

$n_{e}=n_{e0}\left[1-\frac{e\phi}{K_{B}T_{e}(\kappa_{e}-\frac{3}{2})}\right]^{(-\kappa_{e}+\frac{1}{2})},$
$n_{ph}=n_{ph0}\left[1+\frac{e\phi}{K_{B}T_{ph}(\kappa_{p}-\frac{3}{2})}\right]^{(-\kappa_{p}+\frac{1}{2})},$

where $n_{e}$, $n_{ph}$ are the number densities, while $T_{e}$ and $T_{ph}$
are the temperatures of electrons and hot positrons respectively. The real
parameters $\kappa_{e}$ and $\kappa_{p}$ are the super-thermal parameters of
hot electrons and hot positrons respectively.
The normalized basic equations governing the dynamics of positron-acoustic
waves are given in dimensionless variables as follows :

$\frac{\partial n_{pc}}{\partial t}+\frac{\partial(n_{pc}u_{pc})}{\partial
x}=0,$ (1) $\frac{\partial u_{pc}}{\partial t}+u_{pc}\frac{\partial
u_{pc}}{\partial x}=-\frac{\partial\phi}{\partial x},$ (2)
$\frac{\partial^{2}\phi}{\partial
x^{2}}=-n_{pc}-\mu_{ph}\left(1+\frac{\sigma_{1}\phi}{\kappa_{p}-\frac{3}{2}}\right)^{(-\kappa_{p}+\frac{1}{2})}+\mu_{e}\left(1-\frac{\sigma_{2}\phi}{\kappa_{e}-\frac{3}{2}}\right)^{(-\kappa_{e}+\frac{1}{2})}-\mu_{i},$
(3)

where $n_{pc}$ is the cold positron number density normalized by its
equilibrium value $n_{pc0}$, $u_{pc}$ is the cold positron fluid speed
normalized by $C_{pc}=[\frac{K_{B}T_{ef}}{m_{p}}]^{(1/2)}$, $\phi$ is the
electrostatic wave potential normalized by $\frac{K_{B}T_{ef}}{e}$,

$\sigma_{1}=\frac{T_{ef}}{T_{ph}},\sigma_{2}=\frac{T_{ef}}{T_{e}},\mu_{ph}=\frac{n_{ph0}}{n_{pc0}},\mu_{e}=\frac{n_{e0}}{n_{pc0}},\mu_{i}=\frac{n_{i0}}{n_{pc0}},$

$T_{ef}=\frac{T_{e}T_{ph}}{(\mu_{e}T_{ph}+\mu_{ph}T_{e})}$ is the effective
temperature, $K_{B}$ is the Boltzmann constant. $m_{p}$ is the positron mass,
$e$ is the magnitude of the electron charge.
Eq. (2) represents the equation of motion of cold positron. As we assume the
Debye length is $\gg$ gyrofrequency, the effect of all non-electrostatic
fields including the magnetic field is very negligible. The time variable $t$
is normalized by $\omega^{-1}_{pc}=\left[\frac{m_{p}}{4\pi
n_{pc0}e^{2}}\right]^{\frac{1}{2}}$ and the space variable $x$ is normalized
by the Debye length $\lambda_{Dm}=\left[\frac{K_{B}T_{ef}}{4\pi
n_{pc0}e^{2}}\right]^{\frac{1}{2}}$. At equilibrium

$\mu_{ph}=\frac{n_{ph0}}{n_{pc0}}=\frac{n_{e0}}{n_{pc0}}-\frac{n_{i0}}{n_{pc0}}-1=\mu_{e}-\mu_{i}-1.$

Simplifying (3) we get

$\frac{\partial^{2}\phi}{\partial
x^{2}}=-n_{pc}-\mu_{ph}\left[1-a_{1}\sigma_{1}\phi+\frac{a_{2}}{2}\sigma^{2}_{1}\phi^{2}-\frac{a_{3}}{6}\sigma^{3}_{1}\phi^{3}+\dots\right]+\mu_{e}\left[1+b_{1}\sigma_{2}\phi+\frac{b_{2}}{2}\sigma^{2}_{2}\phi^{2}+\frac{b_{3}}{6}\sigma^{3}_{2}\phi^{3}+\dots\right]-\mu_{i},$
(4)

where

$a_{1}=\frac{\kappa_{p}-\frac{1}{2}}{\kappa_{p}-\frac{3}{2}},\;b_{1}=\frac{\kappa_{e}-\frac{1}{2}}{\kappa_{e}-\frac{3}{2}},$
$a_{2}=\frac{(\kappa_{p}-\frac{1}{2})(\kappa_{p}+\frac{1}{2})}{(\kappa_{p}-\frac{3}{2})^{2}},\;b_{2}=\frac{(\kappa_{e}-\frac{1}{2})(\kappa_{e}+\frac{1}{2})}{(\kappa_{e}-\frac{3}{2})^{2}},$
$a_{3}=\frac{(\kappa_{p}-\frac{1}{2})(\kappa_{p}+\frac{1}{2})(\kappa_{p}+\frac{3}{2})}{(\kappa_{p}-\frac{3}{2})^{3}},\;b_{3}=\frac{(\kappa_{e}-\frac{1}{2})(\kappa_{e}+\frac{1}{2})(\kappa_{e}+\frac{3}{2})}{(\kappa_{e}-\frac{3}{2})^{3}}.$

To linearise Eqs. (1)-(3), we consider the dependent variable as the sum of
equilibrium and perturbed parts, so, we can write
$n_{pc}=1+\bar{n_{pc}},u_{pc}=\bar{u_{pc}},\phi=\bar{\phi}.$ Here the values
of parameters at the equilibrium position are given by
$n_{pc}^{(1)}=1,u_{pc}^{(1)}=0,\phi^{(1)}=0.$ Hence we obtain the linearized
forms of Eqs. (1)-(4) are given respectively as follows:

$\frac{\partial\bar{n_{pc}}}{\partial t}+\frac{\partial\bar{u_{pc}}}{\partial
x}=0,$ (5) $\frac{\partial\bar{u_{pc}}}{\partial
t}+\frac{\partial\bar{\phi}}{\partial x}=0,$ (6)
$\frac{\partial^{2}\bar{\phi}}{\partial
x^{2}}=-\bar{n_{pc}}+\mu_{ph}a_{1}\sigma_{1}\bar{\phi}+\mu_{e}b_{1}\sigma_{2}\bar{\phi},$
(7)

To obtain dispersion relation for low-frequency wave, we now assume that the
perturbation is proportional to $e^{i(kx-\omega t)}$ and of the form

$\bar{n_{pc}}=n_{0}e^{i(kx-\omega t)},\bar{u_{pc}}=u_{0}e^{i(kx-\omega
t)},\bar{\phi}=\phi_{0}e^{i(kx-\omega t)}.$

Substituting these values in the above-linearized equations, we get,

$-in_{0}\omega+iku_{0}=0,$ $-iu_{0}\omega+ik\phi_{0}=0,$
$n_{0}-(k^{2}+u_{ph}a_{1}\sigma_{1}+u_{e}b_{1}\sigma_{2})=0.$

Since the above three equations are a system of linear homogeneous equations,
so for nontrivial solutions we have

$\begin{vmatrix}-i\omega&ik&0\\\ 0&-i\omega&ik\\\
1&0&-(k^{2}+u_{ph}a_{1}\sigma_{1}+u_{e}b_{1}\sigma_{2})\end{vmatrix}=0$
$\begin{split}\implies\omega^{2}&=\frac{k^{2}}{(k^{2}+u_{ph}a_{1}\sigma_{1}+u_{e}b_{1}\sigma_{2})}\\\
\implies\omega^{2}&=\frac{k^{2}}{k^{2}+u_{ph}(\frac{\kappa_{p}-\frac{1}{2}}{\kappa_{p}-\frac{3}{2}})\sigma_{1}+u_{e}(\frac{\kappa_{e}-\frac{1}{2}}{\kappa_{e}-\frac{3}{2}})\sigma_{2}},\end{split}$
(8)

this is the dispersion relation.

##  3 Derivation of evolution equations

###  3.1 Korteweg-de Vries (KdV) equation

The KdV equation has been introduced by the following stretched coordinates:

$\xi=\epsilon^{\frac{1}{2}}(x-\lambda t),\tau=\epsilon^{\frac{3}{2}}t,$ (9)

Then the derivatives operators are considered as follows:

$\frac{\partial}{\partial
x}=\epsilon^{\frac{1}{2}}\frac{\partial}{\partial\xi},\frac{\partial}{\partial
t}=\epsilon^{\frac{3}{2}}\frac{\partial}{\partial\tau}-\lambda\epsilon^{\frac{1}{2}}\frac{\partial}{\partial\xi},\frac{\partial^{2}}{\partial
x^{2}}=\epsilon\frac{\partial^{2}}{\partial\xi^{2}}.$

Let the dependent variables are perturbed in the following way:

$n_{pc}=1+\epsilon
n^{(1)}_{pc}+\epsilon^{2}n^{(2)}_{pc}+\epsilon^{3}n^{(3)}_{pc}+\dots,$ (10)
$u_{pc}=0+\epsilon
u^{(1)}_{pc}+\epsilon^{2}u^{(2)}_{pc}+\epsilon^{3}u^{(3)}_{pc}+\dots,$ (11)
$\phi=0+\epsilon\phi^{(1)}+\epsilon^{2}\phi^{(2)}+\epsilon^{3}\phi^{(3)}+\dots.$
(12)

First, we express the Eqs. (1), (2) and (4) in terms of $\xi$ and $\tau$, then
substitute the above expression into Eqs. (1), (2) and (4) . Now equating the
coefficients of each power of $\epsilon$, we get the phase velocity

$\lambda=\left[\frac{1}{\mu_{ph}a_{1}\sigma_{1}+\mu_{e}b_{1}\sigma_{2}}\right]^{1/2}.$
(13)

After some straightforward calculations, we obtained

$\frac{\partial\phi^{(1)}}{\partial\tau}+A_{1}\phi^{(1)}\frac{\partial\phi^{(1)}}{\partial\xi}+B_{1}\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}=0,$
(14)

where
$A_{1}=\frac{1}{2(\mu_{ph}a_{1}\sigma_{1}+\mu_{e}b_{1}\sigma_{2})^{3/2}}[3(\mu_{ph}a_{1}\sigma_{1}+\mu_{e}b_{1}\sigma_{2})^{2}-(\mu_{e}b_{2}\sigma^{2}_{2}-\mu_{ph}a_{2}\sigma^{2}_{1})],$
and
$B_{1}=\frac{\lambda^{3}}{2}=\frac{1}{2}\left[\frac{1}{(\mu_{ph}a_{1}\sigma_{1}+\mu_{e}b_{1}\sigma_{2})^{3/2}}\right].$

###  3.2 Modified Korteweg-de Vries (MKdV) equation

The coefficient of the quadratic nonlinear term
($A_{1}\phi^{(1)}\frac{\partial\phi^{(1)}}{\partial\xi}$) in KdV equation i.e.
$A_{1}$ depends on different parameters, and for a particular parameter set
the value of $A_{1}$ will be $0$, then the equation will become a linear one.
So, to consider nonlinearity in the evolution equation a separate stretching
is needed and accordingly, a new higher-order term i.e. cubic nonlinear term
appears in the evolution equation. For this reason, we employ the following
stretched coordinates for mKdV equation:

$\xi=\epsilon(x-\lambda t),\tau=\epsilon^{3}t.$

Now considering the above perturbed variables (10)-(12), and equating the
lowest power of $\epsilon$ to zero, we obtain the phase velocity as

$\lambda=\left[\frac{1}{(\mu_{ph}a_{1}\sigma_{1}+\mu_{e}b_{1}\sigma_{2})}\right]^{1/2}.$
(15)

Then considering the order of $\epsilon^{3}$, we obtain

$-\frac{1}{2}A_{1}(\phi^{(1)})^{2}=0\implies A_{1}=0.$ (16)

After some calculations, we obtain the mKdV equation in terms of variable
$\phi^{(1)}$,

$\frac{\partial\phi^{(1)}}{\partial\tau}+B_{2}\frac{\partial^{3}\phi^{(1)}}{\partial\xi^{3}}+B_{2}C_{2}(\phi^{(1)})^{2}\frac{\partial\phi^{(1)}}{\partial\xi}=0,$
(17)

where

$B_{2}=\frac{\lambda^{3}}{2}=(B_{1}),C_{2}=\frac{15}{2\lambda^{6}}-(\mu_{ph}\frac{a_{3}}{2}\sigma^{3}_{1}+\mu_{e}\frac{b_{3}}{2}\sigma^{3}_{2}).$

###  3.3 Gardner equation (GE)

The derivation and analysis of the KdV and mKdV equations lay the groundwork
for comprehending how nonlinearity and dispersion interact during wave
propagation. Building on this foundational research, the GE investigates a
wider range of nonlinear phenomena, such as breathers and solitons. So, in
this section, we will derive the GE, and then we will discuss the various
nonlinear structures of the GE.
For $\mu_{ph}$ around its critical value $(\mu_{c})$, $A_{1}=A_{0}$ can be
expressed as:

$A_{0}\approx s\left(\frac{\partial
A_{1}}{\partial\mu_{ph}}\right)_{\mu_{ph}=\mu_{c}}|\mu_{ph}-\mu_{c}|=c_{1}s\epsilon,$

where $|\mu_{ph}-\mu_{c}|$ is a small and dimensionless parameter and can be
taken as the expansion parameter $\epsilon$, i.e.,
$|\mu_{ph}-\mu_{c}|\approx\epsilon$, and $s=1$ for $\mu_{ph}>\mu_{c}$. $s=-1$
for $\mu_{ph}<\mu_{c}$. $c_{1}=\frac{\partial A_{1}}{\partial\mu_{ph}}$ is a
constant depending on the plasma parameters
$\sigma_{1},\sigma_{2},\mu_{ph},\mu_{e},\kappa_{e},\kappa_{p}.$ Now
considering the third order of $\epsilon$ in the Poisson’s equation, we obtain
the Gardner equation (replacing $\phi^{(1)}$ by $\phi$) as:

$\frac{\partial\phi}{\partial\tau}+A_{3}\phi\frac{\partial\phi}{\partial\xi}+C_{3}(\phi)^{2}\frac{\partial\phi}{\partial\xi}+B_{3}\frac{\partial^{3}\phi}{\partial\xi^{3}}=0,$
(18)

where $A_{3}=c_{1}sB_{2},C_{3}=B_{2}C_{2},B_{3}=(B_{2}=B_{1}).$ The Gardner
equation is also called the mixed KdV equation. It contains both $\phi$ term
of KdV and $(\phi)^{2}$ term of mKdV equation.

##  4 Analytical solutions of Gardner equation

To find the soliton solution of the GE (18), we take the transformation

$\phi=2[tan^{-1}(g/f)]_{\xi}.$ (19)

Then the Hirota bilinear forms of (18) are

$(D_{\tau}+B_{3}D^{3}_{\xi})(g.f)=0,$ (20)

and

$3B_{3}D^{2}_{\xi}(f.f+g.g)-A_{3}D_{\xi}(g.f)=0.$ (21)

###  4.1 One-soliton solution

For one-soliton solution we take

$f=1+exp(\theta),$

and

$g=1+d\;exp(\theta),$

where $\theta=p~{}\xi-\Omega\tau-\xi_{0}$; $p$ is the wave number and $\Omega$
is the frequency and $\xi_{0}$ is the initial phase. Using the values of $f$
and $g$ into the Eqs. (20) and (21), we get the dispersion relation
$\Omega=B_{3}p^{3}$ and $d=\frac{A_{3}+6B_{3}p}{A_{3}-6B_{3}p}.$

For $A_{3}>C_{3}~{}p$, the one-soliton solution is,

$\phi(\xi,\tau)=\frac{C_{3}p^{2}}{A_{3}+\sqrt{(A_{3})^{2}+(C_{3})^{2}p^{2}}cosh[p(\xi-
B_{3}p^{2}\tau-\xi_{0})]},$ (22)

where

$exp(p\xi_{0})=\frac{A_{3}-C_{3}p}{\sqrt{(A_{3})^{2}+(C_{3})^{2}p^{2}}},$

and hence the amplitude is

$\frac{C_{3}p^{2}}{A_{3}+\sqrt{(A_{3})^{2}+(C_{3})^{2}p^{2}}}.$

(a) h

(b) h

Figure 1: (a) The 2D graph and (b) The 3D graph of one-soliton solution, by
considering $B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,s=1,p=3$.

Figure 2: (a) The 2D graph and (b) The 3D graph of one-soliton solution, when
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,s=1,p=-3$.

The one-soliton solution of Eq. (18) occurs for $A_{3}<C_{3}~{}p$ and given by

$\phi(\xi,\tau)=-\frac{C_{3}p^{2}}{\sqrt{(A_{3})^{2}+(C_{3})^{2}p^{2}}cosh[p(\xi-
B_{3}p^{2}\tau-\xi_{0})]-A_{3}},$ (23)

where

$exp(p\xi_{0})=\frac{C_{3}p-A_{3}}{\sqrt{(A_{3})^{2}+(C_{3})^{2}p^{2}}}.$

Figure 3: (a) The 2D graph and (b) The 3D graph of one-soliton solution, by
considering $B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,s=1,p=6$.

###  4.2 Two-soliton solutions

For this we take

$f=1+exp(\theta)+exp(\psi)+f_{12}exp(\theta+\psi),$

and

$g=1+d_{1}exp(\theta)+d_{2}exp(\psi)+d_{12}exp(\theta+\psi),$

where $\theta=p_{1}\xi-\Omega_{1}\tau,$ and $\psi=p_{2}\xi-\Omega_{2}\tau;$
$p_{1},p_{2}$ are the wave numbers and $\Omega_{1},\Omega_{2}$ are the
frequencies. Using the values of $f$ and $g$ from Eqs. (20) and (21), we get
the dispersion relation,

$\Omega_{i}=B_{3}p_{i}^{3},~{}i=1,2.$

Now the 2-soliton solutions of equ.(18) is given by

$\phi=[2tan^{-1}(g/f)]_{\xi},$ (24)

where

$\theta=p_{1}\xi-B_{3}p_{1}^{3}\tau,\psi=p_{2}\xi-
B_{3}p_{2}^{3}\tau,d_{n}=\frac{A_{3}+C_{3}p_{n}}{A_{3}-C_{3}p_{n}},n=1,2,$

and

$f_{12}=\frac{(p_{1}-p_{2})^{2}[(A_{3})^{2}-(A_{3})(C_{3})(p_{1}+p_{2})-(C_{3})^{2}p_{1}p_{2}]}{(p_{1}+p_{2})^{2}(A_{3}-C_{3}p_{1})(A_{3}-C_{3}p_{2})},$
$d_{12}=\frac{(p_{1}-p_{2})^{2}[(A_{3})^{2}+(A_{3})(C_{3})(p_{1}+p_{2})-(C_{3})^{2}p_{1}p_{2}]}{(p_{1}+p_{2})^{2}(A_{3}-C_{3}p_{1})(A_{3}-C_{3}p_{2})}$

.

Figure 4: (a) The 2D graph and (b) The 3D graph of two-soliton solutions, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,p_{1}=2,p_{2}=1.5,s=1$.

Figure 5: (a) The 2D graph and (b) The 3D graph of two-soliton solutions, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,p_{1}=-2,p_{2}=1.5,s=1$.

Figure 6: (a) The 2D graph and (b) The 3D graph of two-soliton solutions, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,p_{1}=2,p_{2}=-1.5,s=1$.

Figure 7: (a) The 2D graph and (b) The 3D graph of two-soliton solutions, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,p_{1}=-2,p_{2}=-1.5,s=1$.

###  4.3 Breather Solutions

The breather is obtained from two-soliton solutions by choosing a pair of
complex conjugate wave numbers ($p_{1}=m+in,p_{2}=m-in,$ $m,n$ are real).
After some calculations, we obtain the breather solution of GE, given as
below:

$\phi(\xi,\tau)=[2tan^{-1}(g/f)]_{\xi},$ (25)

where

$g=1-\frac{n^{2}}{m^{2}}\left[\frac{(A_{3})^{2}+2(A_{3})(C_{3})m-(C_{3})^{2}(m^{2}+n^{2})}{(A_{3}-C_{3}m)^{2}+(C_{3})^{2}n^{2}}\right]exp(2\Theta)+2\left[\zeta
cos(\Psi)-\eta sin(\Psi)\right]exp(\Theta),$
$f=1-\frac{n^{2}}{m^{2}}\left[\frac{(A_{3})^{2}-2(A_{3})(C_{3})m-(C_{3})^{2}(m^{2}+n^{2})}{(A_{3}-C_{3}m)^{2}+(C_{3})^{2}n^{2}}\right]exp(2\Theta)+2cos(\Psi)exp(\Theta),$
$\Theta=m[\xi-B_{3}(m^{2}-3n^{2})\tau],\Psi=n[\xi-B_{3}(3m^{2}-n^{2})\tau],$

and

$\zeta=\frac{(A_{3})^{2}-(C_{3})^{2}(m^{2}+n^{2})}{(A_{3}-C_{3}m)^{2}+(C_{3})^{2}n^{2}},\eta=\frac{2(A_{3})(C_{3})n}{(A_{3}-C_{3}m)^{2}+(C_{3})^{2}n^{2}}.$

Figure 8: (a) The 2D graph and (b) The 3D graph of breather solution, by
considering $B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,m=6,n=3,s=1$.

Figure 9: (a) The 2D graph and (b) The 3D graph of breather solution, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,m=-6,n=-3,s=1$.

Figure 10: (a) The 2D graph and (b) The 3D graph of breather solution, by
considering $B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,m=-6,n=3,s=1$.

Figure 11: (a) The 2D graph and (b) The 3D graph of breather solution, by
considering $B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,m=6,n=-3,s=1$.

Figure 12: (a) The 2D graph and (b) The 3D graph of breather solution, by
considering
$B_{3}(=B_{2})=0.396821,C_{2}=0.457408,c_{1}=2.5902,m=-1.7,n=1.5,s=1$.

Figure 13: (a) Potential profile and (b) Field contours of numerical solution
of KdV equation with positive amplitude single soliton as initial profile, by
considering $\mu_{e}=0.7,\mu_{i}=-0.4,\mu_{p}h=0.1,s=1,p=1$.

Figure 14: (a) Potential profile and (b) Field contours of numerical solution
of KdV equation with negative amplitude single soliton as initial profile, by
considering $\mu_{e}=0.7,\mu_{i}=-0.4,\mu_{p}h=0.1,s=1,p=1$.

Figure 15: (a) Potential profile and (b) Field contours of numerical solution
of KdV equation with positive amplitude double solitons as initial profile, by
considering $\mu_{e}=0.7,\mu_{i}=-0.4,\mu_{p}h=0.1,s=1,p=1$.

Figure 16: (a) Potential profile and (b) Field contours of numerical solution
of KdV equation with positive and negative amplitude double solitons as
initial profile, by considering
$\mu_{e}=0.7,\mu_{i}=-0.4,\mu_{p}h=0.1,s=1,p=1$.

Figure 17: (a) Potential profile and (b) Field contours of numerical solution
of mKdV equation with positive amplitude single soliton as initial profile, by
considering $\mu_{e}=0.7,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 18: (a) Potential profile and (b) Field contours of numerical solution
of mKdV equation with negative amplitude single soliton as initial profile, by
considering $\mu_{e}=0.7,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 19: (a) Potential profile and (b) Field contours of numerical solution
of mKdV equation with positive amplitude double solitons as initial profile,
by considering $\mu_{e}=0.7,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 20: (a) Potential profile and (b) Field contours of numerical solution
of mKdV equation with negative amplitude double solitons as initial profile,
by considering $\mu_{e}=0.7,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 21: (a) Potential profile and (b) Field contours of numerical solution
of mKdV equation with positive and negative amplitude double solitons as
initial profile, by considering
$\mu_{e}=0.7,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 22: (a) Potential profile and (b) Field contours of numerical solution
of Gardner equation with positive amplitude single soliton as initial profile,
by considering $\mu_{e}=0.5,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 23: (a) Potential profile and (b) Field contours of numerical solution
of Gardner equation with negative amplitude single soliton as initial profile,
by considering $\mu_{e}=0.5,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 24: (a) Potential profile and (b) Field contours of numerical solution
of Gardner equation with positive amplitude double solitons as initial
profile, by considering $\mu_{e}=0.5,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 25: (a) Potential profile and (b) Field contours of numerical solution
of Gardner equation with negative amplitude double solitons as initial
profile, by considering $\mu_{e}=0.5,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

Figure 26: (a) Potential profile and (b) Field contours of numerical solution
of Gardner equation with positive and negative amplitude double solitons as
initial profile, by considering
$\mu_{e}=0.5,\mu_{i}=-0.6,\mu_{p}h=0.1,s=1,p=1$.

##  5 Results and Summary

###  5.1 Analytical Solutions:

In the previous section, we derived the KdV and mKdV equations, and then for a
critical case, we obtained the Gardner Equation (GE) (18). From the analytical
approach, GE can have either basic 1-soliton solution, basic 2-soliton
solutions, or even a breather solution. These solutions are important from two
points of view. Firstly the solutions are analytically obtained. Hence they
are proof of mathematical soundness. Secondly, they can be related to similar
types of findings in astrophysical and laboratory graphs and interpreted
accordingly. Here we have used the parameter values
$\kappa_{p}=100.41;\kappa_{e}=1.8;\sigma_{1}=0.8242;\sigma_{2}=0.5;\mu_{e}=0.5;\mu_{i}=-0.6$.
For this set of parameters, we get the value of the coefficient of the
quadratic nonlinear term of GE i.e. $A_{1}=1.87039\times 10^{-8},$ which tends
to $0$, so we consider a certain neighborhood around this parameter set, and
get a cubic nonlinear term. For any case, the value of the dispersive
coefficient is $B_{3}(=B_{2}=B_{1})=0.396821.$ From the equilibrium equation
we obtain $\mu_{ph}=0.1$. In the neighborhood where $A_{1}\to 0$, we obtain
the critical value of $\mu_{ph}$ i.e. $\mu_{c}=-3.03917$, so
$\mu_{ph}>\mu_{c}$. Hence $s$ will be $+1$, which appear in
$A_{3}(=c_{1}sB_{2})$. We have seen that the soliton solutions as well as
breathers, do not depend on quadratic coefficient $A_{1}$, it depends on the
expression $c_{1}=\frac{\partial A_{1}}{\partial\mu_{ph}}$ and its value is
$c_{1}=2.5902$.

From the analytical sections we see that the dispersive term for KdV, as well
as mKdV equations, are the same (i.e. $B_{1}=B_{2}=B_{3}$). These are two
types of nonlinearity viz. first-order nonlinearity and second-order
nonlinearity. While the former (1st-order nonlinear term) i.e. $A_{3}$ is a
composite factor given by $A_{3}=c_{1}sB_{2}=c_{1}sB_{1}$. Here
$B_{1}(\text{or}~{}B_{2})$ is the dispersive term of the KdV (or mKdV)
equation. Thus the first-order nonlinearity ($A_{3}$) depends on a variety of
parameters $\sigma_{1},\sigma_{2},\mu_{ph},\mu_{c},\kappa_{e},\kappa_{p}$.

Figures 1(a) & 1(b) and 2 & 2 shows oppositely propagating 1-soliton solution
for the GE. It shows that the compressive soliton simply propagates without
any change in amplitude or width. For phase factor ($p$) being positive, it
propagates as a rarefactive soliton. Comparing figures 2 & 2 and 3 & 3 for
change in value of phase factor ($p$), variable changes appear. The
rarefactive soliton remains unchanged. Next, in Figures 4 to 7, 2-soliton
solutions are investigated. There are two phase factors ($p_{1}~{}\&~{}p_{2}$)
respectively. These figures ($4-7$) provide four combinations of phase
factors. When both $p_{1}~{}\&~{}p_{2}$ are positive (Figures 4 & 4) the
2-soliton approach towards the right merge into a single one, and the
resultant is an overlap of the same and crosses over according to wave speeds.
Figures 5 & 5 and 6 & 6 shows counter propagating solitons in 2-soliton mode.
While the shorter one moves left $(p_{1}<0)$ the larger one moves right
$(p_{2}>0)$. But in this case, the phase difference remains constant over time
since the phase factors are time-independent. Such a situation can be
physically observed in multi-mode optical fiber with a higher standard of
preventive mechanisms towards crosstalk. An interesting occurrence happens
when both $p_{1}~{}\&~{}p_{2}$ are negative (figures7 & 7). If the analogy
were drawn from figure 4 & 4, it would have been like crossing ones, however,
it does not seem to cross over. These remain almost constant phase differences
over time. From the 7 figure, it is seen, the changes are very minor.

We now look for the breather-type solution when periodic modulations are seen
in both space and time. A breather solution is named after its periodic
contraction and expansion. There are several breather in the literature like
Peregrine breathers ([38],[39],[40] ), Ma breathers ([41],[42],[43]),
Akhmediev breathers ([44],[45],[46],[47]) etc. to name a few. Here a new type
of breather has been presented. Starting with complex phase factors $(p_{s})$
i.e. $p_{1}=m+in~{}\&~{}p_{2}=m-in$ where $p_{1}~{}\&~{}p_{2}$ are complex
conjugate suggesting an initial correlation for positive and negative values
of $m~{}\&~{}n$ and $p_{1}~{}\&~{}p_{2}$ have four combinations.
Figure 8& 8 shows when both $m~{}\&~{}n$ are positive. Here the movement of
the breather pulse is right in the space axis suggesting it is a moving
breather. The nature remains almost constant over time.
In figure 9& 9 we see that for both $m<0~{},~{}n<0$, the potential edges of
the breather are not equally positioned i.e. the two boundaries of the
potential wall are not in the same height. This is due to the inertia of the
particles and their response to change due to a propagating wave and the time
taken to rest back in their original position. It seems there is some type of
drag that causes a restitution of energy from the field to the particles.
Figure 10& 10 shows similar nature only the other wall height increases when
compared to figure 9& 9. It is a clear indication that the breathers have a
duration in time for which they exist and after which they break down to a
shock-like structure where there is an infinitely huge potential drop. In such
a region, particles are accelerated due to this huge potential difference. In
figure 11& 11 we show that the breather structure moves towards left. The rest
of nature is the same. An identity inference can be drawn from figure 12& 12.
From the values of $m~{}\&~{}n$ which are very close and oppositely signed,
the breather solution has more of a shock-like features. Such an accelerated
motion of the breather soliton is clearly seen from 12. To sum up this part of
the result we see that starting from the GE three types of solution are
obtained. These solutions help understand the gradual evolution of breather
structures under the framework of GE in such an e-p-i plasmas.

###  5.2 Numerical Solutions:

Thus far we have dealt with analytical solutions and investigated their nature
under different conditions of positive and negative phase factors. Since
analytical solutions for KdV and mKdV equations are available in the
literature; for a change we try with numerical solutions for these. To obtain
the numerical solutions we resort to the ”Fourier Operated fourth-order Runge
Kutta via Exponential Time Difference scheme” (FORKET) code and employ our
Matlab platform to obtain the results. The result part is organized in the
following way; in the first section, we present the figures corresponding to
analytical solutions containing 2-D (field contours) and 3-D (potential
profiles) plots of 1-soliton, 2-soliton and breather solutions for various
parameters. The 3-D figures have time as the third axis whereas the 2-D ones
have it as assuming parameter. From the figures, we see that negative values
of time ($\tau$) have been shown. This has nothing to do with physicality
since we choose the epoch reference somewhere in between. This part of the
numerical scheme was done by two of the authors (SC & CD). Through this
scheme, we have plotted the potential structures ($\phi$) and the fields ($E$)
for KdV, mKdV, and GE. The plots of KdV and mKdV will help us to understand
the intermediate process. The numerical scheme is very much more fine-tuned
than other RK4 schemes. Therefore, with parametric variations, the growth of
KdV, mKdV, and Gardner solitons can be better realized through this tool.

The parameters for numerical plots are given in the caption. First, we study
the evolution of the KdV solitary profile and the electric field as well.
Figure (13) shows the solitonic evolution from the KdV equation. The KdV
soliton propagates towards the right. However, there are secondary modulations
that propagate in the opposite direction. Figure (13) shows the corresponding
electric field in the $\xi-\tau$ plane. Subsequent figures will also show
similar features. A thorough study of this feature suggests two possible
mechanisms.
(i) When positive ions form the ion acoustic wave (IAW), the density pile up
forces the heavier ions to stay back which start small amplitude oscillations
about their mean position thus creating these secondary modulations.
(ii) Alternatively, as ions pile up to form IAW, positrons in the plasma are
repelled back thus forming low amplitude waves which start propagating within
the plasma in the reverse direction to the direction of propagation of IAW
solitons.
When the initial amplitude is taken negative (Figure 14) i.e. ions are
depleted from the soliton resulting in an excess of super-thermal electrons.
These super-thermal electrons create crocodilian back-like secondary humps.
These were not visible in previous cases. However the field nature (figure 14)
is similar except for the sharp intense field that was existing in figure 13.
In figures 15 & 15, two solitonic solutions have been started with. The
initial nature is a hyperbolic secant squared type. The two solitons have
different phase speeds and it is clear from the curved nature of the path of
the soliton that the phase speed is not constant over time, rather a
nonlinearity exists itself in the phase. This curvature however negligible is
proof that the monochromatic concept of solitary wave is not valid completely.
Now if there are two solitons, one compressive and the other rarefactive
(figure 16 & 16), then a train of secondary modulations are seen, these are
deep as well a shallow pockets of particle trapping. The corresponding field
diagram shows the deep potential walls marked with darker shades of blue.
Under certain critical conditions where the nonlinear term of ordinary KdV
evolution is zero, then we take a step forward to obtain the mKdV equation. We
start with a single soliton case and see that both spatially and temporally
modulated solitons are formed and often the single soliton gets deformed and
forms a series of secondary solitons (figure 17). The corresponding field
(figure 17) shows the interaction of primary and secondary solitons. If the
initial structure is that of a rarefactive soliton, the potential deep
gradually becomes less shallow, and secondary solitons are compressive. Here
also the phase does depend on time in a nonlinear fashion (figure 18). The
corresponding fields figure 18 are presented through color code where local
sources and sinks appear either as a continuum or sporadically referring to
small-scale oscillations. The figures (19 & 19) and (20 & 20) we start with
2-soliton solutions of different phase speeds. In the first case, both are
positive compressive solitons whereas in the latter case, both are negative
rarefactive solitons. These two pairs of figures show the intermittent crests
and troughs from which we infer that field lines emerge and converge
throughout the bulk of the plasma. The contour of equipotential surfaces at
the base of potential profiles clearly second the conclusion we drew from..
the field directions. The picturesque structures (figure 21), where
rarefactive and other compressive solitons are taken to be the 2-soliton
solutions of the mKdV equation. Low amplitude solitary profiles grow
progressively showing alterations both in space ($\xi$) and time ($\tau$). The
fields (figure 21) are however not as strong as the previous two cases. This
is visible from the color codes (light green and light blue being
predominant).

Next the numerical solutions for the Gardner equation are given. The spatio-
temporal variations in potential profiles and field contours show the breather
mode to move away from its origin along the phase axis for a single soliton
(figures 22 & 22). If given sufficient time, the breather would appear again
at a different position, where the phase factor matches. If the polarity is
reversed striated cloud-like features appear and it is not any breather mode,
rather a train of successive modulations appears (figures 23 & 23). If two
soliton solutions both compressive (figures 24 & 24) or rarefactive (figures
25 & 25) are studied, positive solitons evolution are different from those of
negative soliton. The possibility of breather modes start appearing. This is
rather interesting since IAWs are formed due to ions, and the positive ones’
evolution is understood. However, with initial negative solitons the secondary
modulations are due to mobile positrons, which are super-thermally Kappa
distributed. This is also reported in the analytic investigation. Lastly, we
study the case for numerical solution, where one compressive and another
rarefactive, regular undulations appear and contours are elongated channels
rather than pockets of crests and troughs in the time axis. Thus through this
numerical technique, we were able to study how the IAW transfers from KdV
solitons to Gardner breathers or solitons in detail.

##  6 Conclusion

The work presented so far contains analytic and numerical studies of KdV,
mKdV, and GE. The derivations are done in detail. Analytic solutions were used
along with numerical solutions. The findings of the analytic solutions of GE
for 1-soliton, 2-solitons, or breathers are presented in detail. Numerically,
all three types of equations were studied. The importance of this work is
twofold. From the mathematical modeling to analytic solutions this work
presents a plethora for future work and can be considered as a theoretician’s
playground. From the applicability part to real physics, the usefulness covers
astrophysics to Laser plasma physics, and even semiconductor plasma where
electron-holes and ions have similar fluid equations. The high energetic
particles often called super-thermal in the literature follow the Kappa
distribution. We have surveyed a lot of literature and found that a non-
Maxwellian distribution is present in many space and laboratory plasma
environments. Depending upon the composition and configurations the physical
problem in a way to suit an astrophysical environment. Going for the
techniques we have employed a sophisticated numerical scheme (ETDRK4) which
does add merit to our work. We have observed that in a certain parameter
region both the quadratic and cubic nonlinearities exist (in GE) and balancing
with dispersion produces interesting soliton structures i.e. breather
structures for a four-component plasma system consisting of immobile positive
ions, mobile cold positrons, and Kappa distributed hot positrons and hot
electrons. No investigation has been done related to this topic. Many
researchers have studied such models but they haven’t investigated the
breather structures in such plasma systems. We first establish the existence
of breathers for GE in a four-component plasma system consisting of immobile
positive ions. mobile cold positrons and Kappa distributed hot positrons and
hot electrons.

## Acknowledgment

Ms.Snehalata Nasipuri is thankful to the Council of Scientific and Industrial
Research (CSIR), India, for providing financial support under the Senior
Research Fellowship (SRF) program (File No: 09/202(0120)/2021-EMR-I).

## Author Contributions

S.N. calculated the analytical part, plotted the results, and wrote the text.
P.C. supervised the whole work. U.N.G. modelled and analysed the results, also
wrote some parts of this article. All authors have discussed to the final
manuscript.

Competing interests and fundings
No funding was received to assist with the preparation of this manuscript.
Also, the author has no financial or proprietary interests in any material
discussed in this article.

Data availability statement
The author confirms that there is no associated data available for the above
research work. Data sharing does not apply to this article as no new data were
created or analyzed in this study.

Declarations Conflict of Interest
The author has no conflict of interest

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