# Heavy ion-acoustic rogue waves in electron-positron multi-ion plasmas

∗N. A. Chowdhury, A. Mannan, M. M. Hasan, and A. A. Mamun  Department of
Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh.
∗Email<EMAIL_ADDRESS>
###### Abstract

The nonlinear propagation of heavy-ion-acoustic (HIA) waves (HIAWs) in a four
component multi-ion plasma (containing inertial heavy negative ions and light
positive ions, as well as inertialess nonextensive electrons and positrons)
has been theoretically investigated. The nonlinear Schrödinger (NLS) equation
is derived by employing the reductive perturbation method. It is found that
the NLS equation leads to the modulational instability (MI) of HIAWs, and to
the formation of HIA rogue waves (HIARWs), which are due to the effects of
nonlinearity and dispersion in the propagation of HIAWs. The conditions for MI
of HIAWs, and the basic properties of the generated HIARWs are identified. It
is observed that the striking features (viz. instability criteria, growth rate
of MI, amplitude and width of HIARWs, etc.) of the HIAWs are significantly
modified by effects of nonextensivity of electrons and positrons, ratio of
light positive ion mass to heavy negative ion mass, ratio of electron number
density to light positive ion number density, and ratio of electron
temperature to positron temperature, etc. The relevancy of our present
investigation to the observations in the space (viz. cometary comae and
earth’s ionosphere) and laboratory (laser plasma interaction experimental
devices) plasmas is pointed out.

##  I Lead Paragraph

A new electron-positron, multi-ion plasma model has been considered to
identify new features (instability criteria, growth rate of MI, amplitude and
width, etc.) of heavy ion-acoustic rogue waves. This rogue waves are
associated with nonlinear propagation of HIAWs in which the inertia (restoring
force) is mainly provided by the heavy negative ions (nonextensive electron
and positron temperatures) and are appeared as the solutions of NLS equation
(derived here by the reductive perturbation method) in unstable parametric
regime. The new striking features of these HIARWs are identified and are found
to be applicable in the space and laboratories plasmas.

##  II Introduction

Over the last few decades, wave dynamics in electron-positron-ion (e-p-i)
plasmas is one of the major research area for the plasma physicists because of
painstaking experimental observational evidence in both space (viz. early
universe Misner1973 , active galactic nuclei Miller1987 , pulsar magnetosphere
Michel1982 , and neutron stars Michel1991 ) and laboratories plasmas (laser-
plasma interaction research Shukla1984 , semiconductor plasmas Shukla1986 ,
and other magnetic confinement systems Surko1990 ) revealed the existence of
nonlinear structures (viz. solitons, envelope solitons, shocks, vortices,
rogue waves etc.) in such kind of multicomponent plasmas. The existence of
negative ions in the cometary comae Chaizy1991  and earth’s ionosphere
Massey1976  is already established. Plasma can contain negative ions along
with positive ions. The negative ions can appear in electronegative plasmas as
a result of elementary processes such as dissociative or nondissociative
electron attachment to neutrals Vladimirov2003 ; Mamun2003 ; Djebli2003 .
Positive and negative ion may coexist simultaneously in neutral beam sources
Bacal1979 , plasma processing reactor Gottscho1986 , and in low-temperature
laboratory experiments Ichiki2002 . Similarly positrons can be generated in
modern laser plasma experiments when ultra-intense laser pulse interacts with
matter Kourakis2006 ; Esfandyari-Khalejahi2006 .

In case of space and laboratory plasmas, all time particles do not follow
Maxwellian distribution (which is a velocity distribution describing the
plasma particles in a thermal equilibrium Maksivonic1997 ; Leubner2004 ;
Christon2012 ). Although, a large number of authors considered that plasma
components are in thermal equilibrium but due to the some external
disturbances (e.g. wave-particle interactions, external force fields present
in natural space plasma environments, and turbulence, etc.) their assumption
is no longer valid. In space and astrophysical environments, the Maxwellian
distribution is no longer exist when the plasma particles move very fast
compared to their thermal velocities. Non-extensive generalization of the
Boltzmann-Gibbs-Shannon entropy, first recognized by Renyi Renyi1955  and
subsequently proposed by Tsallis Tsallis1988  has been obtained a great deal
of interest during last few decades. The nonextensive distribution is
generally denoted by q and in the q-nonextensive framework, the one-
dimensional equilibrium distribution function $f_{s}(v_{s})$ is given
Silva1998  by

$\displaystyle
f_{s}(v_{s})=A_{q}\left[1-(q-1)\frac{m_{s}{v_{s}}^{2}}{2k_{B}T_{s}}\right]^{\frac{1}{q-1}}.\
$

Here the normalization constant is

$\displaystyle
A_{q}=\frac{n_{s0}\Gamma(\frac{1}{1-q})}{\Gamma(\frac{1}{1-q}-\frac{1}{2})}\sqrt{\frac{m_{s}(1-q)}{2\pi
k_{B}T_{s}}},~{}(for-1<q<1),\ $

where $n_{s0}$, $m_{s}$, $v_{s}$, and $T_{s}$ are the equilibrium number
density, mass, thermal speed, and temperature of the energetic particle
species $s$. The thermal speed $v_{s}$ of the nonextensive particle species
$s$ is defined as $(2k_{B}T_{s}/m_{s})$, where $k_{B}$ is the familiar
Boltzmann constant. Nonextensive plasmas are found in cosmological and
astrophysical scenarios (viz. stellar polytropes Plastino1993 , hadronic
matter and quark-gluon plasma Gervino2012 , dark-matter halos Feron2008 ,
Earth’s bow-shock Asbridge1968 , magnetospheres of Jupiter and Saturn
Krimigis1983 , etc.) as well as laboratory applications like nanomaterials,
micro-devices, and micro-structures Vladimirov2004 , etc.

The propagation of wave packets in a nonlinear, dispersive medium is appeared
to the modulation of their wave amplitudes, because of the interaction between
high and low frequency modes, parametric wave coupling, the nonlinear self-
interaction of the carrier wave modes, and such kind of system is governed by
the NLS equation which admits interesting rational solution named rogue waves
(also familiar as freak waves, extreme waves, killer waves, and monster waves)
solution in the modulationally unstable region. The rogue waves were first
observed in ocean Kharif2009 , now it can be seen in plasmas, atmospheric
physics, optics, stock market crashes Yan2010 , and super-fluid helium
Ganshin2008 .

Recently Eslami et al. Eslami2011  investigated the stability of IAWs in
present of q-nonextensive distributed electron and positron plasmas where they
found nonextensive parameter plays a considerable effects to modify stability
conditions of the IAWs. Bacha et al. Bacha2012  analyzed the rogue waves may
be notably affected by electron nonextensivity depending on whether the
parameter $q$ is positive or negative. In multi-ion plasma rogue wave is
examined by Ei-Labany et al. El-Labany2011  they found that within certain
negative ion mass the rogue wave cannot propagate and energy concentration in
rogue wave pulses largely depends on negative ion mass and density. Jannat et
al. Jannat2016  studied Gardner solitons in a multi-ion plasma system with
nonextensive electrons and positrons. The aim of the present work is, by
employing reductive perturbation method a NLS equation is derived to study
rogue waves in multi-ion plasma with nonextensive electron and positron.

The paper is organized in the following fashion: The model equations are
presented in Sec. III. By using reductive perturbation technique, we derived a
NLS equation which governs the slow amplitude evolution in space and time is
given in Sec. IV. The stability of HIAWs and rogue waves are presented in Sec.
V. The conclusion is provided in Sec. VI.

##  III  The Model Equations

We consider an unmagnetized four component plasma system comprising of
inertial light positive ions, heavy negative ions, as well as inertialess
nonextensive electrons and positrons. At equilibrium, the charge neutrality
condition can be expressed as $Z_{1}n_{10}+n_{p0}=Z_{2}n_{20}+n_{e0}$, where
$n_{s0}$ is the unperturbed number densities of the species $s$ ($s=1,2,e,p,$
for light positive ions, heavy negative ions, electrons, and positrons,
respectively) and $Z_{1}~{}(Z_{2})$ is the charge state of positive light ion
(heavy negative ion). The normalized basic equations governing the dynamics of
the IAWs in our considered plasma system are given by

$\displaystyle\frac{\partial n_{1}}{\partial t}+\frac{\partial}{\partial
x}(n_{1}u_{1})=0,$ (1) $\displaystyle\frac{\partial u_{1}}{\partial
t}+u_{1}\frac{\partial u_{1}}{\partial x}=-\frac{\partial\phi}{\partial x},$
(2) $\displaystyle\frac{\partial n_{2}}{\partial t}+\frac{\partial}{\partial
x}(n_{2}u_{2})=0,$ (3) $\displaystyle\frac{\partial u_{2}}{\partial
t}+u_{2}\frac{\partial u_{2}}{\partial x}=\alpha\frac{\partial\phi}{\partial
x},$ (4) $\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=\mu_{e}n_{e}-\mu_{p}n_{p}-n_{1}+(1-\mu_{e}+\mu_{p})n_{2}.$ (5)

The number densities of electrons and positrons following $q-$ distribution
are

$\displaystyle n_{e}=[1+(q-1)\phi]^{\frac{(1+q)}{2(q-1)}},$ $\displaystyle
n_{p}=[1-(q-1)\sigma\phi]^{\frac{(1+q)}{2(q-1)}},$ (6)

where $q$ is the nonextensive parameter describing the degree of
nonextensivity, i.e. $q=1$ corresponds to Maxwellian distribution, whereas
$q<1$ refers to the superextensivity, and the opposite condition $q>1$ refers
to the subextensivity. Substituting Eq. $(6)$ into Eq. $(5)$ and expanding up
to third order, we get

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=\mu_{e}-\mu_{p}-n_{1}+\lambda n_{2}+\gamma_{1}\phi$
$\displaystyle\hskip
31.2982pt+\gamma_{2}\phi^{2}+\gamma_{3}\phi^{3}+\cdot\cdot\cdot\cdot\cdot,$
(7)

where

$\displaystyle\lambda=1-\mu_{e}+\mu_{p},$
$\displaystyle\gamma_{1}=\frac{(\mu_{e}+\mu_{p}\sigma)(q+1)}{2},$
$\displaystyle\gamma_{2}=\frac{(\mu_{e}-\mu_{p}\sigma^{2})(q+1)(3-q)}{8},$
$\displaystyle\gamma_{3}=\frac{(\mu_{e}+\mu_{p}\sigma^{3})(q+1)(q-3)(3q-5)}{48},\
$

and

$\sigma=\frac{T_{e}}{T_{p}}$, $\alpha=\frac{Z_{2}m_{1}}{Z_{1}m_{2}}$,
$\mu_{e}=\frac{n_{e0}}{n_{10}}$, $\mu_{p}=\frac{n_{p0}}{n_{10}}$.

In the above equations, $n_{1}$ ($n_{2}$) is the number density of light
positive ions (heavy negative ions) normalized by its equilibrium value
$n_{10}$ ($n_{20}$); $u_{1}(u_{2})$ is the positive (negative) ion fluid speed
normalized by $C_{1}=(Z_{1}k_{B}T_{e}/m_{1})^{1/2}$, and $\phi$ is the
electrostatic wave potential normalized by $k_{B}T_{e}/e$ (with $e$ being the
magnitude of an electron charge and $k_{B}$ is the Boltzmann constant).
$m_{1}$ $(m_{2})$ is the rest mass of light positive ion (heavy negative ion),
respectively; $T_{e}$ and $T_{p}$ is the temperature of electrons and
positrons respectively. The time and space variables are normalized by
${\omega^{-1}_{pi}}=(m_{1}/4\pi Z^{2}_{1}e^{2}n_{10})^{1/2}$ and
$\lambda_{D1}=(k_{B}T_{e}/4\pi Z_{1}e^{2}n_{10})^{1/2}$, respectively.

|
---|---
Figure 1: Showing the variation of $P/Q$ against $k$ for different values of $q$. (a) For $q=$ positive. (b) For $q=$ negative. Generally all the figures are generated by using these values $\alpha=0.5,\sigma=0.3,\mu_{e}=0.7,$ and $\mu_{p}=0.5$. |
---|---
Figure 2: Variation of $P/Q$ against $k$ for different values of $\mu_{e}$. (a) For $q=1.5$. (b) For $q=-0.7$. |
---|---
Figure 3: Showing the variation of $P/Q$ against $k$ for different values of $\sigma$. (a) For $q=1.5$. (b) For $q=-0.2$. |
---|---
Figure 4: Plot of the of MI growth rate $(\Gamma_{g})$ against ${k_{MI}}$ for different values of $\mu_{p}$. (a) For $q=1.5$. (b) For $q=-0.7$. Along with $k=0.5$ and $\Phi_{0}=0.5$. |
---|---
Figure 5: Variation of $|\Phi|$ against $\xi$ for different values of $\alpha$. (a) For $k=0.2$ and $q=1.5$. (b) For $k=0.5,$ and $q=-0.7$. Along with $\tau=0$. |
---|---
Figure 6: Variation of $|\Phi|$ against $\xi$ for different values of $q$. (a)
For $k=0.2$. (b) For $k=0.5$. Along with $\tau=0$.

##  IV Derivation of the NLS equation

In order to investigate the modulation of the HIAWs in our considered plasma
system, we will derive the NLS equation by employing the reductive
perturbation method. The independent variables are stretched as

$\displaystyle\xi={\epsilon}(x-v_{g}t),~{}~{}~{}\tau={\epsilon}^{2}t,$ (8)

where $v_{g}$ is the group velocity of the wave and $\epsilon$ is a small
parameter. Then we can write a general expression for the dependent variables
as

$\displaystyle
n_{1}=1+\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}n_{1l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ $\displaystyle
u_{1}=\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}u_{1l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ $\displaystyle
n_{2}=1+\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}n_{2l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ $\displaystyle
u_{2}=\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}u_{2l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$
$\displaystyle\phi=\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}\phi_{l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (9)

where k and $\omega$ are real variables representing the carrier wave number
and frequency respectively. Since $n_{1},~{}u_{1},~{}n_{2},~{}u_{2}$, and
$\phi$ must be real, all elements in Eq. $(9)$ satisfy the reality condition
$A_{l}^{(m)}=A_{-l}^{(m)^{*}}$, where the asterisk denotes the complex
conjugate. The derivative operators in the above equations are treated as
follows:

$\displaystyle\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}-\epsilon
v_{g}\frac{\partial}{\partial\xi}+\epsilon^{2}\frac{\partial}{\partial\tau},$
$\displaystyle\frac{\partial}{\partial x}\rightarrow\frac{\partial}{\partial
x}+\epsilon\frac{\partial}{\partial\xi}.$ (10)

Substituting Eqs. $(9)$ and $(10)$ into Eqs. $(1)-(4)$ and $(7)$, then the
first-order approximation $(m=1)$ with the first harmonic $(l=1)$ yields the
following relation

$\displaystyle-i\omega n_{11}^{(1)}+iku_{11}^{(1)}=0,~{}~{}-i\omega
u_{11}^{(1)}+ik\phi_{1}^{(1)}=0,$ $\displaystyle-i\omega
n_{21}^{(1)}+ikn_{21}^{(1)}=0,~{}~{}-ik\alpha\phi_{1}^{(1)}-i\omega
u_{21}^{(1)}=0,$ $\displaystyle
n_{11}^{(1)}-k^{2}\phi_{1}^{(1)}-\gamma_{1}\phi_{1}^{(1)}-\lambda
n_{21}^{(1)}=0.$ (11)

The solution for the first harmonics read as

$\displaystyle
n_{11}^{(1)}=\frac{k^{2}}{\omega^{2}}\phi_{1}^{(1)},~{}~{}~{}~{}~{}~{}~{}~{}u_{11}^{(1)}=\frac{k}{\omega}\phi_{1}^{(1)},$
$\displaystyle n_{21}^{(1)}=-\frac{\alpha
k^{2}}{\omega^{2}}\phi_{1}^{(1)},~{}~{}~{}~{}u_{21}^{(1)}=-\frac{\alpha
k}{\omega}\phi_{1}^{(1)},$ (12)

we thus obtain the dispersion relation for HIAWs

$\displaystyle\omega^{2}=\frac{k^{2}(1+\alpha\lambda)}{k^{2}+\gamma_{1}}.$
(13)

The second-order when $(m=2)$ reduced equations with $(l=1)$ are

$\displaystyle
n_{11}^{(2)}=\frac{k^{2}}{\omega^{2}}\phi_{1}^{(2)}+\frac{2ik(v_{g}k-\omega)}{\omega^{3}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
$\displaystyle
u_{11}^{(2)}=\frac{k}{\omega}\phi_{1}^{(2)}+\frac{i(v_{g}k-\omega)}{\omega^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
$\displaystyle n_{21}^{(2)}=-\frac{\alpha
k^{2}}{\omega^{2}}\phi_{1}^{(2)}-\frac{2ik\alpha(v_{g}k-\omega)}{\omega^{3}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
$\displaystyle u_{21}^{(2)}=-\frac{\alpha
k}{\omega}\phi_{1}^{(2)}-\frac{i\alpha(v_{g}k-\omega)}{\omega^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(14)

whereas the second-order approximation $(m=2)$ with the first harmonic $(l=1)$
gives

$\displaystyle
v_{g}=\frac{\omega(1+\alpha\lambda-\omega^{2})}{k(1+\alpha\lambda)}.$ (15)

The amplitude of the second-order harmonics are found to be proportional to
$|\phi_{1}^{(1)}|^{2}$

$\displaystyle
n_{12}^{(2)}=C_{1}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}~{}~{}~{}n_{10}^{(2)}=C_{6}|\phi_{1}^{(1)}|^{2},$
$\displaystyle
u_{12}^{(2)}=C_{2}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}~{}~{}~{}u_{10}^{(2)}=C_{7}|\phi_{1}^{(1)}|^{2},$
$\displaystyle
n_{22}^{(2)}=C_{3}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}~{}~{}~{}n_{20}^{(2)}=C_{8}|\phi_{1}^{(1)}|^{2},$
$\displaystyle
u_{22}^{(2)}=C_{4}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}~{}~{}~{}u_{20}^{(2)}=C_{9}|\phi_{1}^{(1)}|^{2},$
$\displaystyle\phi_{2}^{(2)}=C_{5}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}~{}~{}~{}\phi_{0}^{(2)}=C_{10}|\phi_{1}^{(1)}|^{2},$
(16)

where

$\displaystyle C_{1}=\frac{3k^{4}+2C_{5}~{}\omega^{2}k^{2}}{2\omega^{4}},$
$\displaystyle C_{2}=\frac{k^{3}+2kC_{5}\omega^{2}}{2\omega^{3}}$
$\displaystyle C_{3}=\frac{3\alpha^{2}k^{4}-2\alpha
k^{2}\omega^{2}C_{5}}{2\omega^{4}},$ $\displaystyle
C_{4}=\frac{\alpha^{2}k^{3}-2\alpha kC_{5}\omega^{2}}{2\omega^{3}},$
$\displaystyle
C_{5}=\frac{2\gamma_{2}\omega^{4}+3\lambda\alpha^{2}k^{4}-3k^{4}}{2\omega^{2}k^{2}-2\omega^{4}(4k^{2}+\gamma_{1})+2\alpha\lambda\omega^{2}k^{2}},$
$\displaystyle C_{6}=\frac{2v_{g}k^{3}+\omega
k^{2}+C_{10}\omega^{3}}{v^{2}_{g}\omega^{3}},$ $\displaystyle
C_{7}=\frac{k^{2}+C_{10}\omega^{2}}{v_{g}\omega^{2}},$ $\displaystyle
C_{8}=\frac{2v_{g}\alpha^{2}k^{3}+\omega\alpha^{2}k^{2}-\alpha
C_{10}\omega^{3}}{v^{2}_{g}\omega^{3}},$ $\displaystyle
C_{9}=\frac{\alpha^{2}k^{2}-\alpha C_{10}\omega^{2}}{v_{g}\omega^{2}},$
$\displaystyle C_{10}=\frac{2\gamma_{2}v^{2}_{g}\omega^{3}+2\lambda
v_{g}\alpha^{2}k^{3}+\omega\lambda\alpha^{2}k^{2}-2v_{g}k^{3}-\omega
k^{2}}{\omega^{3}(1+\alpha\lambda-\gamma_{1}v^{2}_{g})}.\ $

Finally, the third harmonic modes $(m=3)$ and $(l=1)$ and with the help of
Eqs. $(12)-(16)$, give a system of equations, which can be reduced to the
following NLS equation:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,$
(17)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. The dispersion coefficient $P$ is

$\displaystyle
P=\frac{3}{2}\left(\frac{v_{g}}{\omega}-\frac{1}{k}\right)v_{g},\ $

and the nonlinear coefficient $Q$ is given by

$\displaystyle
Q=\frac{\omega^{3}}{2k^{2}(1+\alpha\lambda)}\left[2\gamma_{2}(C_{5}+C_{10})+3\gamma_{3}-\frac{k^{2}(C_{1}+C_{6})}{\omega^{2}}\right.$
$\displaystyle\left.-\frac{2k^{3}(C_{2}+C_{7})}{\omega^{3}}-\frac{\alpha\lambda
k^{2}(C_{3}+C_{8})}{\omega^{2}}-\frac{2\alpha\lambda
k^{3}(C_{4}+C_{9})}{\omega^{3}}\right].\ $

##  V stability and rogue waves

The nonlinear evolution of the HIAWs typically depends on the coefficients of
dispersion term $P$ and nonlinear term $Q$ which are function of the various
plasma parameters such as $\alpha,~{}\sigma,~{}\mu_{e},~{}\mu_{p}$, and $q$.
Thus, these plasma parameters are significantly controlled the stability
conditions of the HIAWs. If $PQ<0$, HIAWs are modulationally stable and for
the case $PQ>0$, HIAWs are modulationally unstable against external
perturbations Kourakis2005 ; Sukla2002 ; Schamel2002 ; Fedele2002  and
simultaneously when $PQ>0$ and ${k_{MI}}<k_{c}$, the growth rate
($\Gamma_{g}$) of MI is given Shalini2015  by

$\displaystyle\Gamma_{g}=|P|~{}{k^{2}_{MI}}\sqrt{\frac{k^{2}_{c}}{k^{2}_{MI}}-1}.$
(18)

Here ${k_{MI}}$ is the perturbation wave number and the critical value of the
wave number of modulation $k_{c}=\sqrt{2Q{|\Phi_{o}|}^{2}/P}$, where
$\Phi_{o}$ is the amplitude of the carrier waves. Hence the maximum value
$\Gamma_{g(max)}$ of $\Gamma_{g}$ is obtained at ${k_{MI}}=k_{c}/\sqrt{2}$ and
is given by $\Gamma_{g(max)}=|Q||\Phi_{0}|^{2}$.

Therefore, we have investigated the stability of the profile by depicting the
ratio of $P/Q$ versus $k$ for different plasma parameters. When $P/Q<0$, HIAWs
are modulationally stable, while $P/Q>0$, HIAWs will be modulationally
unstable against external perturbations. When $P/Q\rightarrow\pm\infty$, the
corresponding value of $k(=k_{c})$ is called critical or threshold wave number
for the onset of MI. Figure $1$ shows the variations of $P/Q$ with $k$ for
possible three ranges $-1<q<0$, $0<q<1$, and $q>1$ of the nonextensive
parameter $q$. It is observed from both Figs. $1(a)$ and $1(b)$ that for small
$k$ there is stable region (HIAWs are modulationally stable and dark envelope
solitons exist), where unstable region (HIAWs are modulationally unstable and
bright envelope solitons exist) is found for large $k$. The critical value
$k_{c}$ is increases (decreases) with an increase in the value of nonextensive
parameter $q$ for the ranges $q>0$ ($q<0$) which can be observed from Figs.
$1(a)$ and $1(b)$, respectively. So the range of nonextensive parameter
(greater than or less than zero) plays a vital role for controlling the
stability of the HIAWs.

We are now investigating the effects of the ratio of electron to light
positive ion number density (via $\mu_{e}$) on stability conditions of HIAWs
which are depicted in Figs. $2(a)$ and $2(b)$. Nonextensive electrons play a
fascinating role to control the stability of HIAWs structure. Within the range
$q>0$, increasing values of electron number density, the critical value
$k_{c}$ is shifted to the higher values that means the instability domain is
occurred at the higher values of $k$ in Fig. $2(a)$. Similar behaviour is
observed for the case $q<0$, the critical value $k_{c}$ is occurred at higher
wave number with increasing of electrons number density in Fig. $2(b)$. So
excess number of electrons are provided more restoring force which maximize
the stability region without depending on the values of nonextensive
parameter.

The effects of temperature parameter (via $\sigma)$ on stable and unstable
regions can be shown from Figs. $3(a)$ and $3(b)$, respectively. The variation
of $P/Q$ with $k$ for different values of $\sigma$ is depicted in Figs. $3(a)$
and $3(b)$ . It is observed that the critical value $(k_{c})$ is shifted
towards higher (lower) values as electron temperature increases for the range
of $q>0$ ($q<0$). The range of nonextensive parameter change the order of the
variation of critical value $(k_{c})$. Thus, electron temperature plays a
crucial role to change the stability of the wave packets.

To highlight the effects of the ratio of positron to light positive ion number
density (via $\mu_{p}$) on the growth rate of MI is depicted in Fig. $4$. The
growth rate of the MI is so much sensitive to change the values of $\mu_{p}$.
It is observed from Figs. $4(a)$ and $4(b)$ that an increase of $\mu_{p}$
value the maximum growth rate is increased (decreased) for $q>0~{}(q<0)$,
respectively. So the nonextensive positrons (via $\mu_{p}$) can be recognized
to enhance or suppress of the MI growth rate of our considered plasma model.

The rogue wave (rational solution) of the NLS Eq. $(17)$ in the unstable
region ($PQ>0$) can be written Ankiewiez2009 ; Abdelwahed2016  as

$\displaystyle\Phi(\xi,\tau)=\sqrt{\frac{2P}{Q}}\left[\frac{4(1+4iP\tau)}{1+16P^{2}\tau^{2}+4\xi^{2}}-1\right]\mbox{exp}(i2P\tau).$
(19)

The solution $(19)$ anticipates the concentration of the HIAWs energy into a
small region that is caused by the nonlinear behavior of the plasma medium
(see Figs. $5$ and $6$). The HIARWs are so much sensitive to change any plasma
parameter. The effect of the ratio of light positive ion mass to heavy
negative ion mass (via $\alpha$) on the HIARWs is shown in Fig. $5(a)$ and
$5(b)$. For the range $q>0$ $(q<0)$, the amplitude and width of the HIARWs are
increased with the increase of $\alpha$ which can be seen in Figs. $5(a)$ and
$5(b)$. However, the HIARWs amplitude decreases with the increase of the heavy
negative ion mass (via $\alpha$). Physically, the increasing of the heavy
negative ions masses lead to dissipate the energy from the system and reduce
the nonlinearity that makes the HIARWs amplitude shorter. Finally, changing
direction of HIARWs amplitude and width are remain invariant, irrespective of
$q$ values. Exact similar fashion can be observed in Fig. $6$, which is
depicted against $q$. For the range $q>0$ $(q<0)$, the amplitude and width of
HIARWs is decreased (decreased) with the increase of $q$ values.

It can be deduced from Figs. $2$ and $5$ that the direction of the variation
of $k_{c}$ and wave profile (hight and width) is totally independent on the
nonextensive parameter ranges. For the ranges of $q>0$ and $q<0$, with an
increase of the ratio of electron to positive light ion number density (via
$\mu_{e}$) and the ratio of light positive ion to heavy negative ion mass (via
$\alpha$), the values of $k_{c}$ and wave profile (hight and width) are
increased which can be observed from Figs. $2$ and $5$, respectively.

##  VI Conclusion

In summary a NLS equation has been derived to describe the small-amplitude
HIAWs in an unmagnetized multi-component plasma consisting of inertial light
positive ions, heavy negative ions, as well as inertialess nonextensive
electrons and positrons. The analysis of the NLS equation reveals the
existence of both stable and unstable region. The critical value $k_{c}$ which
determined the stability/instability region of HIAWs, is totally depend on
various plasma parameters such as nonextensive parameter, electron number
density, and electron temperature. Every plasma parameters plays a vital role
to change the critical value $k_{c}$. The MI of HIAWs can also lead to the
formation of rogue waves in the unstable region in which a large amount of
energy is concentrated in relatively small area in space and time. The hight
and width of rogue waves are greatly depended on the ratio of light positive
ion to heavy negative ion mass which is observed in our present analysis. A
number of observations clearly disclosed the existence of nonextensive
electrons and positrons in various natural space environment and laboratory
plasmas. We are optimistic that the finding of our present investigation
should be useful to understand the nonlinear phenomena in both space (cometary
comae and earth’s ionosphere) and laboratory plasmas (laser plasma) which
containing of inertial light positive ions and heavy negative ions, as well as
inertialess nonextensive electrons and positrons.

## Acknowledgements

N. A. Chowdhury is grateful to the Bangladesh Ministry of Science and
Technology for awarding the National Science and Technology (NST) Fellowship.

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