# Ion-acoustic rogue waves in double pair plasma having non-extensive
particles

S. Jahan∗,1, M.N. Haque∗∗,1, N.A. Chowdhury∗∗∗,2, A. Mannan†,3, and A.A.
Mamun§,1 1Department of Physics, Jahangirnagar University, Savar, Dhaka-1342,
Bangladesh
2 Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
3 Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, Halle,
Germany
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<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The modulational instability (MI) of ion-acoustic (IA) waves (IAWs) and
associated IA rogue waves (IARWs) in double pair plasma containing non-
extensive electrons, iso-thermal positrons, negatively and positively charged
ions have been governed by the standard nonlinear Schrödinger equation (NLSE).
It has been figured out from the numerical study of NLSE that the plasma
system holds modulationally stable (unstable) region in which the dispersive
and nonlinear coefficients of the NLSE have the opposite (same) signs. It is
also found that the fundamental features of IAWs (viz., MI criteria, amplitude
and width of the IARWs, etc.) are rigorously organized by the plasma
parameters such as mass, charge state, and number density of the plasma
components. The existing outcomes of our present study should be helpful for
understanding the nonlinear features of IAWs (viz., MI and IARWs) in both
laboratory and space plasmas.

###### keywords:

Ion-acoustic waves , NLSE , Modulational instability , Rogue waves.

††journal: “Contributions to Plasma Physics”

##  1 Introduction

Double pair plasma (DPP) is characterised as fully ionized gas having
electrons, positron as well as positive and negative ions, and is believed to
exist in astrophysical environments such as Van Allen radiation belt and near
the polar cap of fast rotation neutron stars [1], solar atmosphere [2],
D-region ($H^{+},O_{2}^{-}$) and F-region ($H^{+},H^{-}$) of the earths’s
ionosphere [3], upper region of Titan’s atmosphere [4] and also in laboratory
environments [5, 6, 7, 8, 9, 10]. A number of authors [11, 12, 13] studied
ion-acoustic (IA) waves (IAWs) and associated nonlinear electrostatic
structures namely, solitons, shocks, rogue waves, and double layers in the
DPP.

Maxwellian distribution function is one of the most widely used velocity
distribution functions of particles to describe the dynamics of the iso-
thermal particles. But it has been observed that the characteristics of
majority of particles in the space [14] and laboratory plasma environments
[15] are departed from the Maxwellian distribution. So, to narrate the non-
Maxwellian particles, Renyi [16] first recognized the modification of
Maxwellian distribution, and finally, Tsallis [17] generalized the non-
extensive $q$-distribution. It is noted that the index $q$ in the non-
extensive $q$-distribution characterizes the degree of non-extensivity of the
particles [18]. Shalini et al. [19] studied IAWs in non-extensive plasma
having two-temperature electrons, and observed that the width of the first and
second order IA rogue waves (IARWs) associated with IAWs decreases with
increasing the value of $q$ but the amplitude of the first and second-order
IARWs associated with IAWs is remain constant. Tribeche et al. [20]
investigated electrostatic solitary waves in presence of the non-extensive
electrons, and found that the amplitude of the potential increases with non-
extensive parameter. Hafez and Talukder [21] examined the propagation of the
nonlinear electrostatic waves in a three-component non-extensive plasma having
inertialess non-extensive electrons and positrons, and inertial ions, and
reported that the amplitude of the soliton increases with increasing
temperature of the non-extensive electron.

The investigation of the modulational instability (MI) [22, 23, 24, 25, 26,
27] and associated nonlinear features of wave is one of the most important
research areas for plasma physicists. It is noteworthy that the MI of the wave
is considered to be the primary reason for the formation of massive and
gigantic rogue waves (RWs) [28]. Rogue wave, which is the rational solution of
the standard nonlinear Schrödinger equation (NLSE) [28, 29, 30, 31], is a
short-lived phenomenon which emerges from nowhere and disappears without a
trace [29]. A number of authors have investigated the MI of IAWs by
considering the non-extensive particles [32, 33, 34]. Bains et al. [32]
studied the MI of IAWs in presence of non-extensive electrons, and
demonstrated that the critical wave number ($k_{c}$) at which the instability
sets in increases with the increase in the value of $q$ ($q>0$). Bouzit et al.
[33] investigated the stability conditions of IAWs in presence of non-
extensive non-thermal electrons. Eslami et al. [34] investigated the MI of
IAWs in electron-positron-ion plasma having non-extensive electrons and
positrons, and observed that the $k_{c}$ decreases with $q$ for $q<0$ while
increases with $q$ for $q>0$. To the best knowledge of the authors, no attempt
has been made to investigate the MI of the IAWs and associated IARWs in a
four-component plasma containing inertial positively and negatively charged
ions, and inertialess non-extensive electrons, and iso-thermal positrons.
Therefore, it is a rational fascination to examine the influence of non-
extensive electrons and iso-thermal positrons on the MI of IAWs and associated
IARWs in a four-component DPP.

The manuscript is organized as the following pattern: The model equations are
presented in Sec. 2. The derivation of the NLSE is shown in Sec. 3. The
stability of IAWs is provided in Sec. 4. The IARWs is demonstrated in Sec. 5.
Finally, a conclusion is given in Sec. 6.

##  2 Model Equations

We consider the propagation of IAWs in a collisionless, fully ionized,
unmagnetized plasma system consisting of warm negative ions, symbolized by
$n_{-i}$ (charge $q_{-i}=-Z_{-i}e$; mass $m_{-i}$), warm positive ions,
denoted by $n_{+i}$ (charge $q_{+i}=Z_{+i}e$; mass $m_{+i}$), non-extensive
$q$-distributed electrons, identified by $n_{e}$ (charge $q_{e}=-e$; mass
$m_{e}$), and iso-thermal positrons, expressed by $n_{p}$ (charge $q_{p}=+e$;
mass $m_{p}$); where $Z_{-i}$ ($Z_{+i}$) is the charge state of the negatively
(positively) charged ion, and $e$ being the magnitude of the charge of the
electron. The charge neutrality condition of our present model can be written
as $n_{p0}+Z_{+i}n_{+i0}=n_{e0}+Z_{-i}n_{-i0}$. Now, the normalized equations
can be given in the following form

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

where $n_{-i}$, $n_{+i}$, $n_{e}$, and $n_{p}$ are normalized by $n_{-i0}$,
$n_{+i0}$, $n_{e0}$, and $n_{p0}$, respectively; $u_{-i}$ and $u_{+i}$
indicate the negatively and positively charged ion fluid, respectively,
normalized by the IA wave speed $C_{-i}=(Z_{-i}k_{B}T_{e}/m_{-i})^{1/2}$ (with
$k_{B}$ being the Boltzmann constant and $T_{e}$ being the temperature of the
electron); $\phi$ denoted as the electrostatic wave potential, normalized by
$k_{B}T_{e}/e$; the time and space variables are, respectively normalized by
$\omega_{-ip}^{-1}=(m_{-i}/4\pi e^{2}Z_{-i}^{2}n_{-i0})^{1/2}$ and
$\lambda_{-i}=(k_{B}T_{e}/4\pi e^{2}Z_{-i}n_{-i0})^{1/2}$. The pressure term
of the ion can be represented as $P_{-i}=P_{-i0}(N_{-i}/n_{-i0})^{\gamma}$
with $P_{-i0}=n_{-i0}k_{B}T_{-i}$; where $P_{-i0}$ ($T_{-i}$) being the
equilibrium pressure (temperature) of the negatively charged ion, and
$P_{+i}=P_{+i0}(N_{+i}/n_{+i0})^{\gamma}$ with $P_{+i0}=n_{+i0}k_{B}T_{+i}$;
where $P_{+i0}$ ($T_{+i}$) being the equilibrium pressure (temperature) of the
positively charged ion, respectively, and $\gamma=(N+2)/N$ (where $N$
recognized as the degree of freedom and for one-dimensional case $N=1$, so
$\gamma=3$). Other parameters can be defined as
$\lambda_{1}=3T_{-i}/Z_{-i}T_{e}$, $\lambda_{2}=Z_{+i}m_{-i}/Z_{-i}m_{+i}$,
$\lambda_{3}=3T_{+i}m_{-i}/Z_{-i}m_{+i}T_{e}$,
$\lambda_{4}=n_{e0}/Z_{-i}n_{-i0}$, and
$\lambda_{5}=Z_{+i}n_{+i0}/Z_{-i}n_{-i0}$. Now, the number densities of the
non-extensive $q-$distributed [17, 35] electron and iso-thermally distributed
[36, 37] positron can be represented by the following normalized equations

$\displaystyle n_{e}=[1+(q-1)\phi]^{\frac{q+1}{2(q-1)}},$ (6) $\displaystyle
n_{p}=\mbox{exp}(-\lambda_{6}\phi),\ $ (7)

where $\lambda_{6}=T_{e}/T_{p}$ (with $T_{p}$ being the temperature of the
iso-thermally distributed positron and $T_{e}>T_{p}$). The parameter $q$,
generally known as entropic index which quantifies the degree of non-
extensivity. It is noteworthy that when $q=1$, the entropy reduces to standard
Maxwell-Boltzmann distribution. On the other hand, in the limits $q>0$
($q<0$), the entropy shows sub-extensivity (super-extensivity). Now, by
substituting Eqs. $(6)$ and $(7)$ into Eq. $(5)$ and expanding up to third
order in $\phi$, we can draw up as

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}+1+\lambda_{5}n_{+i}=\lambda_{5}+n_{-i}+M_{1}\phi+M_{2}\phi^{2}+M_{3}\phi^{3}+\cdot\cdot\cdot,\
$ (8)

where

$\displaystyle
M_{1}=[\lambda_{4}(q+1)+2\lambda_{6}(1+\lambda_{4}-\lambda_{5})]/2,$
$\displaystyle
M_{2}=[\lambda_{4}(q+1)(3-q)-4\lambda_{6}^{2}(1+\lambda_{4}-\lambda_{5})]/8,$
$\displaystyle
M_{3}=[\lambda_{4}(q+1)(q-3)(3q-5)+8\lambda_{6}^{3}(1+\lambda_{4}-\lambda_{5})]/48.\
$

It is noted that the terms containing $M_{1}$, $M_{2}$, and $M_{3}$ in Eq. (8)
are due to the contribution of the non-extensive $q$-distributed electrons and
iso-thermal positrons.

##  3 Derivation of the NLSE

To study the MI of the IAWs, first we want to construct the NLSE by employing
the reductive perturbation method. In that case, the stretched co-ordinates
can be written in the following fashion [35, 38, 39, 40, 41, 42]

$\displaystyle\xi=\epsilon(x-v_{g}t),$ (9) $\displaystyle\tau=\epsilon^{2}t,\
$ (10)

where $v_{g}$ is the group speed and $\epsilon$ is a small parameter. After
that the dependent variables can be represented as [35, 38, 39, 40, 41, 42]

$\displaystyle\Pi(x,t)=\Pi_{0}+\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}\Pi_{l}^{(m)}(\xi,\tau)M,\
$ (11)

where
$\Pi_{il}^{(m)}=[n_{-il}^{m},u_{-il}^{m},n_{+il}^{m},u_{+il}^{m},\phi_{il}^{m}]$,
$\Pi_{0}=[1,0,1,0,0]^{T}$, $M=\mbox{exp}[il(kx-\omega t)]$, and $k~{}(\omega)$
is real variables representing the carrier wave number (frequency). The
derivative operators can be showed as

$\displaystyle\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}-\epsilon
v_{g}\frac{\partial}{\partial\xi}+\epsilon^{2}\frac{\partial}{\partial\tau},$
(12) $\displaystyle\frac{\partial}{\partial
x}\rightarrow\frac{\partial}{\partial
x}+\epsilon\frac{\partial}{\partial\xi}.\ $ (13)

Now, by substituting Eqs. (9)$-$(13) into Eqs. (1)$-$(4), and (8), and
collecting power term of $\epsilon$, the first order ($m=1$ with $l=1$)
shortened equations can be presented as

$\displaystyle n_{-i1}^{(1)}=\frac{k^{2}}{S}\phi_{1}^{(1)},$ (14)
$\displaystyle u_{-i1}^{(1)}=\frac{k\omega}{S}\phi_{1}^{(1)},$ (15)
$\displaystyle n_{+i1}^{(1)}=\frac{\lambda_{2}k^{2}}{A}\phi_{1}^{(1)},$ (16)
$\displaystyle u_{+i1}^{(1)}=\frac{\lambda_{2}\omega k}{A}\phi_{1}^{(1)},\ $
(17)

where $S=\lambda_{1}k^{2}-\omega^{2}$ and $A=\omega^{2}-\lambda_{3}k^{2}$.
These equations provide the dispersion relation of IAWs in the following form

$\displaystyle\omega^{2}=\frac{I\pm\sqrt{I^{2}-4UJ}}{2U},$ (18)

where
$I=(\lambda_{1}k^{2}+\lambda_{3}k^{2}+\lambda_{1}M_{1}+\lambda_{3}M_{1}+\lambda_{2}\lambda_{5}+1)$,
$U=(k^{2}+M_{1})/k^{2}$, and
$J=k^{2}(\lambda_{1}\lambda_{3}k^{2}+\lambda_{1}\lambda_{3}M_{1}+\lambda_{3}+\lambda_{1}\lambda_{2}\lambda_{5})$.
However, to obtain the real and positive values of $\omega$, the conditions
$I^{2}>4UJ$ must be maintained. It is noted that the positive and negative
signs in Eq. (18) resembled to the fast ($\omega_{f}$) and slow ($\omega_{s}$)
IA modes, respectively. The second order ($m=2$ with $l=1$) equations and with
the compatibility condition, we can be written the group speed of the IAWs

$\displaystyle
v_{g}=\frac{(L+\lambda_{1}A^{2}k^{2}+A^{2}\omega^{2}-2A^{2}S^{2}-SA^{2})}{2\omega
k(A^{2}+\lambda_{2}\lambda_{5}S^{2})},\ $ (19)

where
$L=\lambda_{2}\lambda_{3}\lambda_{5}k^{2}S^{2}+\lambda_{2}\lambda_{5}\omega^{2}S^{2}+\lambda_{2}\lambda_{5}AS^{2}$.
Now, the co-efficient of $\epsilon$ (when $m=2$ with $l=2$) yield the second-
order harmonic amplitudes are found to be proportional to
$|\phi_{1}^{(1)}|^{2}$

$\displaystyle n_{-i2}^{(2)}=M_{4}|\phi_{1}^{(1)}|^{2},$ (20) $\displaystyle
u_{-i2}^{(2)}=M_{5}|\phi_{1}^{(1)}|^{2},$ (21) $\displaystyle
n_{+i2}^{(2)}=M_{6}|\phi_{1}^{(1)}|^{2},$ (22) $\displaystyle
u_{+i2}^{(2)}=M_{7}|\phi_{1}^{(1)}|^{2},$ (23)
$\displaystyle\phi_{2}^{(2)}=M_{8}|\phi_{1}^{(1)}|^{2},\ $ (24)

where

$\displaystyle
M_{4}=\frac{2M_{8}k^{2}S^{2}-3\omega^{2}k^{4}-\lambda_{1}k^{6}}{2S^{3}},$
$\displaystyle M_{5}=\frac{\omega M_{4}S^{2}-\omega k^{4}}{kS^{2}},$
$\displaystyle
M_{6}=\frac{2\lambda_{2}M_{8}A^{2}k^{2}+3\lambda_{2}^{2}\omega^{2}k^{4}+\lambda_{3}\lambda_{2}^{2}k^{6}}{2A^{3}},$
$\displaystyle M_{7}=\frac{\omega
M_{6}A^{2}-\omega\lambda_{2}^{2}k^{4}}{kA^{2}},$ $\displaystyle
M_{8}=\frac{2M_{2}A^{3}S^{3}-3\lambda_{5}\lambda_{2}^{2}\omega^{2}S^{3}k^{4}-M_{9}}{M_{10}},$
$\displaystyle
M_{9}=3\omega^{2}A^{3}k^{4}+\lambda_{3}\lambda_{5}\lambda_{2}^{2}S^{3}k^{6}+\lambda_{1}A^{3}k^{6},$
$\displaystyle M_{10}=2A^{2}S^{2}(S\lambda_{2}\lambda_{5}k^{2}-4k^{2}AS-
M_{1}AS-Ak^{2}).\ $

Next, consider the image for $m=3$ with $l=0$ and $m=2$ with $l=0$, which
margined the zeroth harmonic modes. In such a way we can get the following
results

$\displaystyle n_{-i0}^{(2)}=M_{11}|\phi_{1}^{(1)}|^{2},$ (25) $\displaystyle
u_{-i0}^{(2)}=M_{12}|\phi_{1}^{(1)}|^{2},$ (26) $\displaystyle
n_{+i0}^{(2)}=M_{13}|\phi_{1}^{(1)}|^{2},$ (27) $\displaystyle
u_{+i0}^{(2)}=M_{14}|\phi_{1}^{(1)}|^{2},$ (28)
$\displaystyle\phi_{0}^{(2)}=M_{15}|\phi_{1}^{(1)}|^{2},\ $ (29)

where

$\displaystyle M_{11}=\frac{2\omega
v_{g}k^{3}+\lambda_{1}k^{4}+k^{2}\omega^{2}-M_{15}S^{2}}{S^{2}(v_{g}^{2}-\lambda_{1})},$
$\displaystyle M_{12}=\frac{M_{11}v_{g}S^{2}-2\omega k^{3}}{S^{2}},$
$\displaystyle
M_{13}=\frac{\lambda_{2}M_{15}A^{2}+\lambda_{3}\lambda_{2}^{2}k^{4}+\lambda_{2}^{2}\omega^{2}k^{2}+2\omega
v_{g}\lambda_{2}^{2}k^{3}}{A^{2}(v_{g}^{2}-\lambda_{3})},$ $\displaystyle
M_{14}=\frac{M_{13}v_{g}A^{2}-2\omega\lambda_{2}^{2}k^{3}}{A^{2}},$
$\displaystyle M_{15}=\frac{A^{2}(v_{g}^{2}-\lambda_{3})(2\omega
v_{g}k^{3}+\lambda_{1}k^{4}+\omega^{2}k^{2})+M_{16}}{A^{2}S^{2}v_{g}^{2}+M_{17}},$
$\displaystyle
M_{16}=2M_{2}A^{2}S^{2}(v_{g}^{2}-\lambda_{1})(v_{g}^{2}-\lambda_{3})-(v_{g}^{2}-\lambda_{1})$
$\displaystyle S^{2}(2\lambda_{5}\omega
v_{g}\lambda_{2}^{2}k^{3}+\lambda_{3}\lambda_{5}\lambda_{2}^{2}k^{4}+\lambda_{5}\lambda_{2}^{2}\omega^{2}k^{2}),$
$\displaystyle
M_{17}=A^{2}S^{2}[\lambda_{2}\lambda_{5}(v_{g}^{2}-\lambda_{1})-\lambda_{3}-M_{1}(v_{g}^{2}-\lambda_{1})(v_{g}^{2}-\lambda_{3})].\
$

Lastly, the third-order harmonic modes ($m=3$ with $l=1$), with the assistance
of Eqs. (14)$-$(29), represent a complete set of equations, which can be
transformed to the NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,\
$ (30)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. In Eq. (30), $P$ is the dispersion
co-efficient, which can be written as

$\displaystyle
P=\frac{4\lambda_{2}\lambda_{5}\omega^{2}k^{2}v_{g}^{2}S^{3}+4\lambda_{1}\omega
v_{g}A^{3}k^{3}+4kv_{g}\omega^{3}A^{3}+M_{18}}{2\omega
ASk^{2}(A^{2}+\lambda_{2}\lambda_{5}S^{2})},\ $ $\displaystyle
M_{18}=2\lambda_{2}\lambda_{3}\lambda_{5}\omega^{2}k^{2}S^{3}+\lambda_{2}\lambda_{5}\lambda_{3}^{2}S^{3}k^{4}+\lambda_{1}Sk^{2}A^{3}$
$\displaystyle+\lambda_{2}\lambda_{5}S^{3}\omega^{4}+\lambda_{2}\lambda_{3}\lambda_{5}Ak^{2}S^{3}-4\lambda_{2}\lambda_{5}kv_{g}\omega^{3}S^{3}$
$\displaystyle-4\lambda_{2}\lambda_{3}\lambda_{5}\omega
v_{g}k^{3}S^{3}-4\omega^{2}k^{2}v_{g}^{2}A^{3}-\lambda_{1}^{2}A^{3}k^{4}$
$\displaystyle-2\lambda_{1}\omega^{2}k^{2}A^{3}-Sk^{2}v_{g}^{2}A^{3}-A^{3}\omega^{4}$
$\displaystyle-\lambda_{2}\lambda_{5}Ak^{2}v_{g}^{2}S^{3}-A^{3}S^{3},\ $

and $Q$ is the nonlinear co-efficient, which can be written as

$\displaystyle
Q=\frac{3M_{3}A^{2}S^{2}+2M_{2}M_{8}A^{2}S^{2}+2M_{2}M_{15}A^{2}S^{2}-M_{19}}{2\omega
k^{2}(A^{2}+\lambda_{2}\lambda_{5}S^{2})},\ $

where

$\displaystyle M_{19}=2\omega M_{5}A^{2}k^{3}+2\lambda_{2}\lambda_{5}\omega
M_{7}S^{2}k^{3}+2\omega M_{12}A^{2}k^{3}$
$\displaystyle+2\lambda_{2}\lambda_{5}\omega
M_{14}S^{2}k^{3}+\lambda_{1}M_{4}A^{2}k^{4}+M_{4}\omega^{2}A^{2}k^{2}$
$\displaystyle+\lambda_{2}\lambda_{3}\lambda_{5}M_{6}S^{2}k^{4}+\lambda_{2}\lambda_{5}M_{6}\omega^{2}k^{2}S^{2}+M_{11}\omega^{2}A^{2}k^{2}$
$\displaystyle+\lambda_{1}M_{11}A^{2}k^{4}+\lambda_{2}\lambda_{5}M_{13}\omega^{2}k^{2}S^{2}+\lambda_{2}\lambda_{3}\lambda_{5}M_{13}S^{2}k^{4}$

It may be noted here that both $P$ and $Q$ are directly depend on different
parameters namely $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$,
$\lambda_{5}$, $\lambda_{6}$, $q$, and are indirectly depend on mass, number
density, temperature, and charge state of the different plasma components.

Figure 1: The variation of $P/Q$ with $k$ for different values of
$\lambda_{2}$ when $\lambda_{1}=0.007$, $\lambda_{3}=0.07$, $\lambda_{4}=1.8$,
$\lambda_{5}=1.5$, $\lambda_{6}=1.7$, $q=1.4$, and $\omega\equiv\omega_{f}$.
Figure 2: The variation of $P/Q$ with $k$ for different values of
$\lambda_{2}$ when $\lambda_{1}=0.007$, $\lambda_{3}=0.07$, $\lambda_{4}=1.8$,
$\lambda_{5}=1.5$, $\lambda_{6}=1.7$, $q=1.4$, and $\omega\equiv\omega_{s}$.
Figure 3: The variation of $P/Q$ with $k$ for different values of
$\lambda_{4}$ when $\lambda_{1}=0.007$, $\lambda_{2}=1.2$, $\lambda_{3}=0.07$,
$\lambda_{5}=1.5$, $\lambda_{6}=1.7$, $q=1.4$, and $\omega\equiv\omega_{f}$.
Figure 4: The variation of $P/Q$ with $k$ for different values of $q$ when
$\lambda_{1}=0.007$, $\lambda_{2}=1.2$, $\lambda_{3}=0.07$, $\lambda_{4}=1.8$,
$\lambda_{5}=1.5$, $\lambda_{6}=1.7$, and $\omega\equiv\omega_{f}$. Figure 5:
The variation of $P/Q$ with $k$ for different values of $q$ when
$\lambda_{1}=0.007$, $\lambda_{2}=1.2$, $\lambda_{3}=0.07$, $\lambda_{4}=1.8$,
$\lambda_{5}=1.5$, $\lambda_{6}=1.7$, and $\omega\equiv\omega_{f}$. Figure 6:
The variation of $|\Phi|$ with $\xi$ for different values of $\lambda_{5}$
when $\lambda_{1}=0.007$, $\lambda_{2}=1.2$, $\lambda_{3}=0.07$,
$\lambda_{4}=1.8$, $\lambda_{6}=1.7$, $\tau=0$, $k=0.4$, and
$\omega\equiv\omega_{f}$. Figure 7: The variation of $|\Phi|$ with $\xi$ for
different values of $\lambda_{6}$ when $\lambda_{1}=0.007$, $\lambda_{2}=1.2$,
$\lambda_{3}=0.07$, $\lambda_{4}=1.8$, $\lambda_{5}=1.5$, $\tau=0$, $k=0.4$,
and $\omega\equiv\omega_{f}$.

##  4 Stability of IAWs

The propagation of IAWs is modulationaly stable when $P$ and $Q$ have opposite
sign (i.e., $P/Q<0$), and is modulationally unstable when both $P$ and $Q$
have same sign (i.e., $P/Q>0$). The point at which the transition of $P/Q$
curve intersect with the $k$-axis in the “$P/Q$ versus $k$” graph is known as
threshold or critical wave number $k~{}(=k_{c})$. Under consideration of fast
and slow IA modes, we have depicted the $P/Q$ versus $k$ curve for different
values of $\lambda_{2}$ in Figs. 1 and 2, respectively, and it can be seen
from these two figures that (a) both stable and unstable parametric regimes
are allowed by the plasma system; (b) the $k_{c}$ increases (decreases) with
increasing in the value of the negatively (positively) charged ion mass for a
fixed value of their charge states; (c) the stable parametric regime increases
(decreases) with positive (negative) ion charge state when their masses remain
constant.

The effect of electron number density on the stability condition of IAWs can
be understood by plotting $P/Q$ with $k$ for different values of $\lambda_{4}$
in Fig. 3. It is easy to demonstrate from this figure that the stable
(unstable) parametric regime of IAWs increases (decreases) with the increase
in the value of the equilibrium electron number density. The impact of sub-
extensivity and super-extensivity of electrons on the stability condition of
IAWs can be seen in Figs. 4 and 5, respectively. It is obvious from these two
figures that the sub-extensive property of the electrons allows the IAWs to be
stable for large wave number while the super-extensive property of the
electrons allows the IAWs to be stable for small wave number.

##  5 Rogue Waves

The first-order rogue wave solution of the NLSE can be written as [28]

$\displaystyle\Phi(\xi,\tau)=\sqrt{\frac{2P}{Q}}\Big{[}\frac{4+16i\tau
P}{1+4\xi^{2}+16\tau^{2}P^{2}}-1\Big{]}\mbox{exp}(2i\tau P).\ $ (31)

It is worthy to mention that the first-order rogue wave solution of the NLSE
indicates that a considerable amount of IAWs energy is condensed into a very
small domain in DPP. The effect of the number density and charge state of both
positively and negatively charged ions on the amplitude and width of the IARWs
can be observed from Fig. 6. It is noted that an increase in the number
density of positive (negative) ions tend to enhance (decrease) both the
amplitude and width of the IARWs in the modulationally unstable parametric
regime ($P/Q>0$) for a constant value of positive and negative ions charge
state. The physics of this result is that an increase in the value of positive
(negative) ion number density tend to increase (decrease) the nonlinearity as
well as amplitude and width of the IARWs. The nature of IARWs may also be
affected by the electron and positron temperature which can be observed in
Fig. 7. This figure reveals that an increase in the value of the electron
(positron) temperature would make the amplitude and width of the IARWs
associated with IAWs smaller (taller). The physics behind this result is that
the nonlinearity of the plasma medium as well as height and width of the IARWs
decreases (increases) with electron (positron) temperature.

##  6 Conclusion

We have scrutinized the MI of IAWs and associated IARWs in a four-component
DPP having inertial positive and negative ions and inertialess non-extensive
electrons and iso-thermal positrons by deriving a standard NLSE. It is noted
that all of the plasma components in a DPP medium play a vigorous role in the
stability criteria of the IAWs. However, the essence of our findings can be
summarized as follows:

  1. 1.

Under consideration of fast and slow mode, both stable and unstable parametric
regimes of IAWs can be observed.

  2. 2.

The sub-extensive property of the electrons allows the IAWs to be stable for
large wave number while the super-extensive property of the electrons allows
the IAWs to be stable for small wave number.

  3. 3.

An increase in the value of positive (negative) ion number density tend to
increase (decrease) the nonlinearity as well as amplitude and width of the
IARWs.

  4. 4.

The nonlinearity of the plasma medium as well as height and width of the IARWs
decreases (increases) with electron (positron) temperature.

The implications of our present investigation will be useful in understanding
the process of MI of IAWs and associated IARWs in both laboratory plasma
[viz., processing reactors [6], semiconductor industry [7], neutral beam
sources [5], fullerene ($C^{+}_{60},C^{-}_{60}$) [13], and in intense laser
fields [8]] and astrophysical environments [viz., Van Allen radiation belt and
near the polar cap of fast rotation neutron stars [1], solar atmosphere [2],
D-region ($H^{+},O_{2}^{-}$) and F-region ($H^{+},H^{-}$) of the earths’s
ionosphere [3], upper region of Titan’s atmosphere [4], etc.].

## Acknowledgement

S. Jahan gratefully acknowledge NST (National Science and Technology)
Fellowship for their financial support.

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