# Dust-acoustic rogue waves in non-thermal plasmas

S. K. Paul∗1, N. A. Chowdhury∗∗,2,3, A. Mannan†,2,4, and A. A. Mamun‡,2
1Department of Basic Sciences and Humanities, University of Asia Pacific,
Farmgate, Dhaka-1205, Bangladesh
2Department of Physics, Jahangirnagar University, Savar, Dhaka-1342,
Bangladesh
3Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
4Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, D-06099
Halle, Germany
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The nonlinear propagation of dust-acoustic (DA) waves (DAWs) and associated DA
rogue waves (DARWs), which are governed by the nonlinear Schrödinger equation,
is theoretically investigated in a four component plasma medium containing
inertial warm negatively charged dust grains and inertialess non-thermal
distributed electrons as well as iso-thermal positrons and ions. The
modulationally stable and unstable parametric regimes of DAWs are numerically
studied for the plasma parameters. Furthermore, the effects of temperature
ratios of ion-to-electron and ion-to-positron, and the number density of ion
and dust grains on the DARWs are investigated. It is observed that the
physical parameters play a very crucial role in the formation of DARWs. These
results may be useful in understanding the electrostatic excitations in dusty
plasmas in space and laboratory situations.

###### keywords:

NLSE , Modulational instability , Rogue waves.

††journal: “Pramana-Journal of Physics”

##  1 Introduction

The research regarding the propagation of nonlinear electrostatic perturbation
in a four component electron-positron-ion-dust (FCEPID) plasma medium
(FCEPIDPM) has received a great attention among the plasma physicists due to
the existence of FCEPID not only in space plasmas, viz., the hot spots on dust
rings in the galactic centre [1, 2, 3, 4], auroral zone [2], around pulsars
[3, 5], interstellar medium [6], Milky Way [6], accretion disks near neutron
stars [6], Jupiter’s magnetosphere [7] but also in laboratory experiments. A
number of authors have studied dust-acoustic (DA) waves (DAWs) [3, 4, 5],
dust-ion-acoustic waves (DIAWs) [1, 2], and associated nonlinear structures in
four component dusty plasma. The presence of positron drastically changes the
mechanism of the formation and propagation of nonlinear electrostatic
structures, viz., shock, solitons [2], super-solitons [1, 7], and double
layers (DLs) [2], etc.

The existence of the non-thermal particles (viz., electrons [1, 2, 6, 7],
positrons [1, 2], and ions, etc.), which rigorously change the dynamics of the
plasma medium and the mechanism of the formation of various electrostatic
pulses, in space plasmas has confirmed by the Viking [8] and Freja satellites
[9]. Cairns et al. [10] investigated the electrostatic solitary structures in
a non-thermal plasma, and found that both negative and positive density
perturbations can exist in the presence of non-thermal electrons. Paul et al.
[1] examined that the existence of non-thermal electrons and positrons in a
FCEPID model supports the solitary waves of both polarities. Banerjee and
Maitra [2] studied DIAWs, i.e., solitons and DLs, in a FCEPIDPM having non-
thermal electrons and positrons and observed that the positive potential can
not exist after a particular value of $\alpha$.

The rogue waves (RWs), which appear due to the modulational instability (MI)
of the carrier waves, have been observed in different branch of sciences,
viz., Oceanography [11], Biology, Optics [12], Finance [13], Plasma physics
[14, 15, 16, 17], and are also governed by the nonlinear Schrödinger equation
(NLSE) [14, 15, 16, 17]. Chowdhury et al. [15] examined the stability
conditions for the positron-acoustic waves (PAWs) in a multi-component plasma
medium, and found that the critical wave number ($k_{c}$), which indicates the
stable and unstable parametric regimes of PAWs, reduces with the increase in
the value of $\alpha$. Kourakis and Shukla [16] studied the MI of the DIAWs in
a three component plasma medium and showed that the existence of the negative
dust grains reduces the value of $k_{c}$. Rahman et al. [17] reported the
stable and unstable domains in a dusty plasma medium having non-thermal plasma
species, and found that the amplitude of the DA RWs (DARWs) increases with
non-thermality of the plasma species.

Recently, Saberian et al. [4] studied the DA solitons and DLs in a FCEPIDPM.
Esfandyari-Kalejahi et al. [5] investigated DA solitary (DASWs) in a multi-
component dusty plasma in presence of non-thermal electrons, and found that
the amplitude of the DASWs increases with charge state of the negative dust
grains. Jehan et al. [3] considered an unmagnetized FCEPIDPM having negatively
charged massive inertial dust grains, inertialess electrons, positrons, and
ions for examining DASWs, and observed that the existence of positrons and
it’s temperature can change the sign of the nonlinear coefficient of the
governing equation. The aim of the present paper is to further significant
amount of extension of Jehan et al. [3] work by studying the MI of the DAWs in
an unmagnetized FCEPIDPM having non-thermal electrons featuring Cairns’
distribution, and also examine the nonlinear properties of DARWs.

The manuscript is organized as follows: The basic governing equations of our
plasma model are presented in Sec. 2. The MI and RWs are presented in Sec. 4.
Results and discussion are provided in Sec. 5. The conclusion is provided in
Sec. 6.

##  2 Governing Equations

We consider the propagation of DAWs in an unmagnetized collisionless FCEPIDPM
consisting of inertial warm negatively charged massive dust grains (mass
$m_{d}$; charge $q_{d}=-Z_{d}e$), and inertialess non-thermal Cairns’
distributed electrons (mass $m_{e}$; charge $q_{e}=-e$) as well as iso-thermal
positrons (mass $m_{p}$; charge $q_{p}=+e$) and ions (mass $m_{i}$; charge
$q_{i}=+Z_{i}e$), where $Z_{d}$ ($Z_{i}$) is the number of electron (proton)
residing on a negatively (positively) charged massive dust grains (ions).
Overall, the charge neutrality condition for our plasma model can be written
as $n_{e0}+Z_{d}n_{d0}=n_{p0}+Z_{i}n_{i0}$. Now, the basic set of normalized
equations can be written as

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

where $n_{d}$ is the number density of warm dust grains normalized by its
equilibrium value $n_{d0}$; $u_{d}$ is the dust fluid speed normalized by the
DA wave speed $C_{d}=(Z_{d}k_{B}T_{i}/m_{d})^{1/2}$ (with $T_{i}$ being the
ion temperature, $m_{d}$ being the dust grain mass, and $k_{B}$ being the
Boltzmann constant); $\phi$ is the electrostatic wave potential normalized by
$k_{B}T_{i}/e$ (with $e$ being the magnitude of single electron charge); the
time and space variables are normalized by $\omega_{pd}^{-1}=(m_{d}/4\pi
Z_{d}^{2}e^{2}n_{d0})^{1/2}$ and $\lambda_{Dd}=(k_{B}T_{i}/4\pi
Z_{d}n_{d0}e^{2})^{1/2}$, respectively; $P_{d}=P_{d0}(N_{d}/n_{d0})^{\gamma}$
[with $P_{d0}$ being the equilibrium adiabatic pressure of the dust, and
$\gamma=(N+2)/N$, where $N$ is the degree of freedom and for one-dimensional
case, $N=1$ then $\gamma=3$]; $P_{d0}=n_{d0}k_{B}T_{d}$ (with $T_{d}$ being
the temperatures of the warm dust grains); and other plasma parameters are
considered as $\sigma_{1}=3T_{d}/Z_{d}T_{i}$, $\mu_{e}=n_{e0}/Z_{d}n_{d0}$,
and $\mu_{i}=Z_{i}n_{i0}/Z_{d}n_{d0}$. Now, the expression for the number
density of non-thermal electrons following the Cairns’ distribution [1, 10]
can be written as

$\displaystyle
n_{e}=[1-\beta\sigma_{2}\phi+\beta\sigma_{2}^{2}\phi^{2}]\exp(\sigma_{2}\phi),\
$ (4)

where $\sigma_{2}=T_{i}/T_{e}$ ($T_{e}$ being the temperature of the iso-
thermal electron), and $\beta=4\alpha/(1+3\alpha)$ with $\alpha$ being the
parameter determining the fast particles present in our plasma model. Now, the
expression for the number density of iso-thermal positrons following the
Maxwellian distribution can be written as [3]

$\displaystyle n_{p}=\exp(-\sigma_{3}\phi),\ $ (5)

where $\sigma_{3}=T_{i}/T_{p}$ ($T_{p}$ being the temperature of the iso-
thermal positrons). Now, the expression for the number density of iso-thermal
ions following the Maxwellian distribution can be written as [3]

$\displaystyle n_{i}=\exp(-\phi).\ $ (6)

Now, by substituting Eqs. (4)-(6) in Eq. (3), and expanding up to third order
of $\phi$, we get

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}+1=n_{d}+S_{1}\phi+S_{2}\phi^{2}+S_{3}\phi^{3}+...\ $ (7)

where

$\displaystyle
S_{1}=(\sigma_{2}-\beta\sigma_{2}+\sigma_{3})\mu_{e}+(1-\sigma_{3})\mu_{i}+\sigma_{3},$
$\displaystyle
S_{2}=[{(\sigma_{2}^{2}-\sigma_{3}^{2})\mu_{e}+(\sigma_{3}^{2}-1)\mu_{i}-\sigma_{3}^{2}}]/{2},$
$\displaystyle
S_{3}=[{(3\beta+1)\mu_{e}\sigma_{2}^{3}+\mu_{e}\sigma_{3}^{3}+(1-\sigma_{3}^{3})\mu_{i}+\sigma_{3}^{3}}]/{6}.\
$

The terms containing $S_{1}$, $S_{2}$, and $S_{3}$ in the right hand side of
Eq. (7) are the contribution of inertialess electrons, positrons, and ions. We
note that Eqs. (1), (2), and (7) now represent the basis set of normalized
equations to describe the nonlinear dynamics of the DAWs, and associated DARWs
in an unmagnetized FCEPIDPM under consideration.

##  3 Derivation of the NLSE

To study the MI of the DAWs, we want to derive the NLSE by employing the
reductive perturbation method, and for that case, first we can write the
stretched co-ordinates in the form [14, 15, 16, 17]

$\displaystyle\xi={\epsilon}(x-v_{g}t),$ (8)
$\displaystyle\tau={\epsilon}^{2}t,\ $ (9)

where $v_{g}$ is the group speed and $\epsilon$ is a small parameter. Then we
can write the dependent variables as

$\displaystyle
n_{d}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{dl}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (10) $\displaystyle
u_{d}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{dl}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (11)
$\displaystyle\phi=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}\phi_{l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (12)

where $k$ and $\omega$ are real variables representing the carrier wave number
and frequency, respectively. The derivative operators can be written as [14,
15, 16, 17]

$\displaystyle\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}-\epsilon
v_{g}\frac{\partial}{\partial\xi}+\epsilon^{2}\frac{\partial}{\partial\tau},$
(13) $\displaystyle\frac{\partial}{\partial
x}\rightarrow\frac{\partial}{\partial
x}+\epsilon\frac{\partial}{\partial\xi}.$ (14)

Now, by substituting Eqs. (8)-(14) into Eqs. (1)- (2), and (7), and collecting
the terms containing $\epsilon$, the first order ($m=1$ with $l=1$) reduced
equations can be written as

$\displaystyle n_{d1}^{(1)}=\frac{k^{2}}{S}\phi_{1}^{(1)},$ (15)
$\displaystyle u_{d1}^{(1)}=\frac{k\omega}{S}\phi_{1}^{(1)},$ (16)
$\displaystyle n_{d1}^{(1)}=-k^{2}\phi_{1}^{(1)}-S_{1}\phi_{1}^{(1)},\ $ (17)

where $S=\sigma_{1}k^{2}-\omega^{2}$. Hence these relations provide the
dispersion relation of DAWs

$\displaystyle\omega^{2}=\sigma_{1}k^{2}+\frac{k^{2}}{S_{1}+k^{2}}.\ $ (18)

The second-order ($m=2$ with $l=1$) equations are given by

$\displaystyle
n_{d1}^{(2)}=\frac{k^{2}}{S}\phi_{1}^{(2)}-\frac{2ik\omega(v_{g}k-\omega)}{S^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(19) $\displaystyle
u_{d1}^{(2)}=\frac{k\omega}{S}\phi_{1}^{(2)}-\frac{i(v_{g}k-\omega)(\omega^{2}+\sigma_{1}k^{2})}{S^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},\
$ (20)

with the compatibility condition

$\displaystyle v_{g}=\frac{\partial\omega}{\partial
k}=\frac{\omega^{2}-S^{2}}{k\omega}.\ $ (21)

The coefficients of $\epsilon$ for $m=2$ and $l=2$ provide the second order
harmonic amplitudes which are found to be proportional to
$|\phi_{1}^{(1)}|^{2}$

$\displaystyle n_{d2}^{(2)}=S_{4}|\phi_{1}^{(1)}|^{2},$ (22) $\displaystyle
u_{d2}^{(2)}=S_{5}|\phi_{1}^{(1)}|^{2},$ (23)
$\displaystyle\phi_{2}^{(2)}=S_{6}|\phi_{1}^{(1)}|^{2},\ $ (24)

where

$\displaystyle
S_{4}=\frac{2S_{6}k^{2}S^{2}-\sigma_{1}k^{6}-3\omega^{2}k^{4}}{2S^{3}},$
$\displaystyle S_{5}=\frac{S_{4}\omega S^{2}-\omega k^{4}}{kS^{2}},$
$\displaystyle
S_{6}=\frac{\sigma_{1}k^{6}+3\omega^{2}k^{4}-2S_{2}S^{3}}{6k^{2}S^{3}}.\ $

Now, we consider the expression for ($m=3$ with $l=0$) and ($m=2$ with $l=0$),
which leads the zeroth harmonic modes. Thus, we obtain

$\displaystyle n_{d0}^{(2)}=S_{7}|\phi_{1}^{(1)}|^{2},$ (25) $\displaystyle
u_{d0}^{(2)}=S_{8}|\phi_{1}^{(1)}|^{2},$ (26)
$\displaystyle\phi_{0}^{(2)}=S_{9}|\phi_{1}^{(1)}|^{2},\ $ (27)

where

$\displaystyle S_{7}=\frac{S_{9}S^{2}-\sigma_{1}k^{4}-\omega^{2}k^{2}-2\omega
v_{g}k^{3}}{S^{2}(\sigma_{1}-v^{2}_{g})},$ $\displaystyle
S_{8}=\frac{S_{7}v_{g}S^{2}-2\omega k^{3}}{S^{2}},$ $\displaystyle
S_{9}=\frac{2\omega
v_{g}k^{3}+\sigma_{1}k^{4}+\omega^{2}k^{2}-2S_{2}S^{2}(\sigma_{1}-v^{2}_{g})}{S^{2}(1-S_{1}v^{2}_{g}+\sigma_{1}S_{1})}.\
$

Finally, the third harmonic modes ($m=3$) and ($l=1$), with the help of
(15)-(27), give a set of equations, which can be reduced to the following
NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,$
(28)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. In Eq. (28), the dispersion
coefficient $P$ is as

$\displaystyle P=\frac{F_{1}(v_{g}k-\omega)-S^{3}}{2\omega k^{2}S}.\ $

where
$F_{1}=\omega^{3}+3\omega\sigma_{1}k^{2}-3v_{g}k\omega^{2}-v_{g}\sigma_{1}k^{3}$,
and also in Eq. (28), the nonlinear coefficient $Q$ is as

$\displaystyle Q=\frac{2S_{2}S^{2}(S_{9}+S_{6})+3S_{3}S^{2}-F_{2}}{2\omega
k^{2}}.\ $

Figure 1: Plot of $P/Q$ vs $k$ for different values of $\alpha$ along with
$\mu_{e}=0.5$, $\mu_{i}=0.7$, $\sigma_{1}=0.003$, $\sigma_{2}=0.4$, and
$\sigma_{3}=0.5$. Figure 2: Plot of $P/Q$ vs $k$ for different values of
$\mu_{e}$ along with $\alpha=0.6$, $\mu_{i}=0.7$, $\sigma_{1}=0.003$,
$\sigma_{2}=0.4$, and $\sigma_{3}=0.5$.

where $F_{2}=(k^{4}\sigma_{1}+\omega^{2}k^{2})(S_{4}+S_{7})+2\omega
k^{3}(S_{5}+S_{8})$. The space and time evolution of the DAWs in an
unmagnetized FCEPIDPM are directly governed by the coefficients $P$ and $Q$,
and indirectly governed by different plasma parameters such as $\alpha$,
$\mu_{i}$, $\mu_{e}$, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$, and $k$. Thus,
these plasma parameters significantly affect the stability conditions of the
DAWs in an unmagnetized FCEPIDPM.

##  4 Modulational instability and Rogue waves

The stable and unstable parametric regimes of DAWs are organized by the sign
of $P$ and $Q$ of Eq. (28) [14, 15, 16, 17, 18, 19, 20]. When $P$ and $Q$ have
same sign (i.e., $P/Q>0$), the evolution of the DAWs amplitude is
modulationally unstable in presence of the external perturbations. On the
other hand, when $P$ and $Q$ have opposite sign (i.e., $P/Q<0$), the DAWs are
modulationally stable in presence of the external perturbations. The plot of
$P/Q$ against $k$ yields stable and unstable parametric regimes of the DAWs.
The point, at which the transition of $P/Q$ curve intersect with $k$-axis, is
known as threshold or critical wave number $k$ ($=k_{c}$) [18, 19, 20, 21, 22,
23, 24, 25].

Figure 3: Plot of $P/Q$ vs $k$ for different values of $\sigma_{2}$, along
with $\alpha=0.6$, $\mu_{e}=0.5$, $\mu_{i}=0.7$, $\sigma_{1}=0.003$, and
$\sigma_{3}=0.5$.

The governing equation for highly energetic DARWs in the modulationally
unstable parametric regime ($P/Q>0$) can be written as [26, 27, 28, 29, 30]

$\displaystyle\Phi(\xi,\tau)=\sqrt{\frac{2P}{Q}}\left[\frac{4(1+4iP\tau)}{1+16P^{2}\tau^{2}+4\xi^{2}}-1\right]\mbox{exp}(2iP\tau).$
(29)

Figure 4: Plot of $|\Phi|$ vs $\xi$ for different values of $\mu_{i}$, along
with $k=1.7$, $\tau=0$, $\alpha=0.6$ $\mu_{e}=0.5$, $\sigma_{1}=0.003$,
$\sigma_{2}=0.4$, and $\sigma_{3}=0.5$.

Equation (29) describes that a large amount of wave energy, which causes due
to the nonlinear characteristics of the plasma medium, is localized into a
comparatively small area in space.

##  5 Results and discussion

We have numerically analyzed the stable and unstable parametric regimes of
DAWs in Figs. 1-3. The effects of non-thermality of the electrons in
organizing the stable and unstable parametric regimes of the DAWs can be seen
from Fig. 1 which indicates that the $k_{c}$ as well as modulationally stable
parametric regime of DAWs decreases with the increase in the value of
$\alpha$, and this result is a good agreement with the result of Ref. [15].
So, the DAWs become modulationally unstable for small values of $k$ for excess
non-thermality of the plasma species.

Figure 2 indicates that the presence of the non-thermal electrons and
negatively charged warm dust grains can significantly modify the stability
conditions of DAWs in an unmagnetized FCEPIDPM. It is clear from this figure
that (a) when $\mu_{e}=0.3$, $0.5$, and $0.7$ then the corresponding $k_{c}$
value is $k_{c}\equiv 1.4$ (dotted blue curve), $k_{c}\equiv 1.5$ (dashed
green curve), and $k_{c}\equiv 1.6$ (solid red curve); (b) the critical value
increases with the increase in the value of the $\mu_{e}$; (c) the
modulationally stable (unstable) parametric regime of DAWs increases with an
increase in the value of electron (dust) number density for constant value of
the charge state of negative dust grains (via $\mu_{e}$).

Figure 5: Plot of $|\Phi|$ vs $\xi$ for different values of $\sigma_{3}$ along
with $k=1.7$, $\tau=0$, $\alpha=0.6$ $\mu_{e}=0.5$, $\mu_{i}=0.7$,
$\sigma_{1}=0.003$, and $\sigma_{2}=0.4$.

The effects of ion to electron temperature (via $\sigma_{2}$) on the stable
and unstable parametric regimes can be observed in Fig. 3, and it is clear
from this figure that (a) the $k_{c}$ increases with $\sigma_{2}$; (b) the
modulationally stable parametric regime increases with ion temperature while
decreases with electron temperature. So, the temperature of the electron and
ion plays an opposite role in recognizing the modulationally stable and
unstable parametric regimes of DAWs in an unmagnetized FCEPIDPM.

We have numerically analyzed Eq. (29) in Figs. 4 and 5 to understand the
nonlinear property of the FCEPIDPM as well as the mechanism of the formation
of DARWs associated with DAWs in the modulationally unstable parametric
regime. Figure 4 indicates that (a) the amplitude and width of the DARWs
increase with an increase in the value of $\mu_{i}$; (b) the nonlinearity,
which organizes the shape of the DARWs, of the plasma medium increases with an
increase in the value of ion number density whereas the nonlinearity of the
plasma medium decreases with dust number density when their charge state
remain constant. Figure 5 indicates the temperature effects of ion and
electron (via $\sigma_{3}$) on the generation of DARWs in FCEPIDPM. The ion
(positron) temperature enhances (suppresses) both amplitude and width of the
DARWs associated with DAWs in the modulationally unstable parametric regime.

##  6 Conclusion

In this study, we have performed a nonlinear analysis of DAWs in an
unmagnetized FCEPIDPM consisting of inertial negatively charged warm dust
grains, and inertialess non-thermal Cairns’ distributed electrons as well as
iso-thermal positrons and ions. The evolution of DAWs is governed by the
standard NLSE, and the coefficients $P$ and $Q$ of NLSE can recognize the
modulationally stable and unstable parametric regimes of DAWs in the presence
of the external perturbation. It is observed that the DAWs become unstable for
small values of $k$ for excess non-thermality (via $\alpha$) of the plasma
species, and the stable (unstable) parametric regimes of the DAWs increases
with the increase in the value of electron (dust) number density at constant
value of the dust charge state. The ion (positron) temperature enhances
(suppresses) both amplitude and width of the DARWs. Finally, these results may
be applicable in understanding the conditions of the MI of DAWs and associated
DARWs in astrophysical environments (viz., the hot spots on dust rings in the
galactic centre [1, 2, 3, 4], auroral zone [2], around pulsars [3, 5],
interstellar medium [6], Milky Way [6], accretion disks near neutron stars
[6], and Jupiter’s magnetosphere [7], etc) and laboratory devices.

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