# Dust-acoustic envelope solitons in super-thermal plasmas

A. A. Noman∗,1, N. A. Chowdhury∗∗,1, A. Mannan1,2, and A. A. Mamun1 1
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
2 Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, Halle,
Germany
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The modulational instability (MI) of the dust-acoustic waves (DAWs) in an
electron-positron-ion-dust plasma (containing super-thermal electrons,
positrons and ions along with negatively charged adiabatic dust grains) is
investigated by the analysis of the nonlinear Schrödinger equation (NLSE). To
derive the NLSE, the reductive perturbation method has been employed. Two
different parametric regions for stable and unstable DAWs are observed. The
presence of super-thermal electrons, positrons and ions significantly modifies
both the stable and unstable regions. The critical wave number $k_{c}$ (at
which modulational instability sets in) depends on the super-thermal electron,
positron, and ion, and adiabatic dust concentrations.

###### keywords:

Dust-acoustic waves , modulational instability , envelope solitons.

††journal: “Contributions to Plasma Physics”

##  1 Introduction

The existence of positrons in a four components dusty plasma medium (DPM)
along with electron-ion-dust in space plasmas (viz., supernova environments
[1, 2, 3], in the hot spots on dust rings in the galactic centre [4],
interstellar clouds [1], pulsar magnetosphere [2], Milky Way [4], and
accretion disks near neutron stars [4], etc.) as well as laboratory plasmas
[1] has encouraged researchers to investigate the propagation of electrostatic
perturbation. Plasma physicists have encountered with dust-acoustic (DA) waves
(DAWs) [1, 2, 3], dust-ion-acoustic (DIA) waves (DIAWs) [4], DIA cnoidal waves
(DIA-CWs) [5] as well as their associated nonlinear structures such as shock,
solitary [1, 2], and double layers (DLs) [6, 7] to manifest the nonlinear
intrinsic properties of an electron-positron-ion-dust (EPID) plasma medium in
presence of highly energetic positrons [1, 2, 3, 4].

The empirical results have shown the signature of the fast particle in the
space [9, 5, 6, 7, 8] and laboratory DPM. These fast particles are governed by
the super-thermal/$\kappa$-distribution which first introduced by the
Vasyliunas [9]. The $\kappa$-distribution overlaps with the Maxwellian
distribution when the super-thermal parameter $\kappa$ in
$\kappa$-distribution travels to infinite (i.e., $\kappa\rightarrow\infty$).
Saini and Sethi [5] examined DIA-CWs in a four components DPM, and found that
the width and amplitude of the DIA-CWs decrease with increasing the value of
$\kappa$. Alam et al. [6] investigated solitary waves (SWs) and DLs regarding
DIAWs in a DPM in presence of super-thermal electrons, and observed that the
amplitude of the negative DLs potential increases with super-thermal parameter
$\kappa$. Dutta and Goswami [7] demonstrated DIA DLs in a four components DPM,
and highlighted that the amplitude of the DLs rigourously depends on the
number density of the negative dust as well as super-thermality of the
electrons. Ghosh et al. [8] reported DA SWs (DASWs) in a multi-component
plasmas, and found that the amplitude of the SWs decreases with $\kappa$.

A world of plasma would be a boring place since the nonlinear interaction are
at the heart of the working of nature. The nonlinear Schrödinger equation
(NLSE) and associated envelope solitonic solutions are the most interesting
theory which can solve the puzzle of nature. A number of authors have
investigated the modulational instability (MI) of the carrier waves by
employing the NLSE [10, 11, 12, 13, 14, 15, 16, 17]. Chowdhury et al. [10]
examined the existence of the bright and dark envelope solitons in a quantum
plasma medium, and observed that the thickness of the bright and dark envelope
solitons is crucially affected by the variation of the plasma parameters while
the hight of envelope solitons remains constant. Ahmed et al. [11]
investigated the MI of the ion-acoustic waves in presence of
$\kappa$-distributed electrons and positrons, and found that the critical wave
number ($k_{c}$) decreases with increasing the value of $\kappa$. Bains et al.
[12] analyzed the MI of the DAWs in three components DPM. Saini and Kourakis
[13] considered three components DPM having super-thermal plasma species, and
investigated the MI of DAWs as well as formation of the envelope solitons, and
also found that super-thermality of the plasma species leads to narrower
bright envelope solitons.

Recently, a number of authors [1, 2, 3] have considered a dusty plasma system
and studied the nonlinear DASWs. However, to the best knowledge of authors, no
attempt has been made on the nonlinear behaviour of DA wave packets and their
MI in EPID plasma system. Therefore, in our present work, a four components
plasma system (containing super-thermal electrons, positrons, and ions, and
adiabatic negatively charged dust grains) has been considered to investigate
the nonlinear DAWs and their MI as well as bright and dark envelope solitons.

The outline of the paper is as follows: The governing equations describing our
plasma model are presented in Section 2. Derivation of NLSE is devoted in
Section 3. The modulational instability of DAWs is given in Section 4.
Envelope solitons are mentioned in Section 5. A brief conclusion is, finally,
provided in Section 6.

##  2 Governing Equations

We consider a four components plasma model consisting of inertial adiabatic
dust grains, inertialess super-thermal electrons, positrons, and ions. At
equilibrium, the quasi-neutrality condition can be expressed as
$n_{e0}+Z_{d}n_{d0}=n_{p0}+Z_{i}n_{i0}$, where $n_{e0}$, $n_{d0}$, $n_{p0}$,
and $n_{i0}$ are, respectively, the equilibrium number densities of electrons,
adiabatic dust grains, positrons, and ions. The normalized governing equations
to study the DAWs can be written as:

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

where $n_{d}$ is the adiabatic dust grains number density 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 adiabatic dust grains); and other plasma parameters
are considered as $\sigma_{1}=3T_{d}/Z_{d}T_{i}$,
$\sigma_{2}=n_{p0}/Z_{d}n_{d0}$, and $\sigma_{3}=Z_{i}n_{i0}/Z_{d}n_{d0}$. The
expression for the number density of electrons, positrons, and ions following
the $\kappa$-distribution [11] can be expressed, respectively, as

$\displaystyle
n_{e}=\left[1-\frac{\sigma_{4}\phi}{(\kappa_{e}-3/2)}\right]^{-\kappa+\frac{1}{2}},$
(4) $\displaystyle
n_{p}=\left[1+\frac{\sigma_{5}\phi}{(\kappa_{p}-3/2)}\right]^{-\kappa+\frac{1}{2}},$
(5) $\displaystyle
n_{i}=\left[1+\frac{\phi}{(\kappa_{i}-3/2)}\right]^{-\kappa+\frac{1}{2}},\ $
(6)

where $\sigma_{4}=T_{i}/T_{e}$ and $\sigma_{5}=T_{i}/T_{p}$. The super-
thermality of electrons, positrons, and ions is, respectively, represented by
the $\kappa_{e}$, $\kappa_{p}$, and $\kappa_{i}$. We consider
$\kappa_{e}=\kappa_{p}=\kappa_{i}=\kappa$ for numerical analysis. Now, by
substituting (4)-(6) into (3), and expanding up to third order of $\phi$, we
get

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

where

$\displaystyle
H_{1}=(\sigma_{2}+\sigma_{3}-1)\gamma_{1}\sigma_{4}+\gamma_{1}\sigma_{2}\sigma_{5}+\gamma_{1}\sigma_{3},~{}~{}~{}~{}~{}H_{2}=(\sigma_{2}+\sigma_{3}-1)\gamma_{2}\sigma_{4}^{2}-\gamma_{2}\sigma_{2}\sigma_{5}^{2}-\gamma_{2}\sigma_{3},$
$\displaystyle
H_{3}=(\sigma_{2}+\sigma_{3}-1)\gamma_{3}\sigma_{4}^{3}+\gamma_{3}\sigma_{2}\sigma_{5}^{3}+\gamma_{3}\sigma_{3},~{}~{}~{}~{}~{}\gamma_{1}=(2\kappa-1)/(2\kappa-3),$
$\displaystyle\gamma_{2}=[(2\kappa-1)(2\kappa+1)]/2(2\kappa-3)^{2},~{}~{}~{}~{}~{}\gamma_{3}=[(2\kappa-1)(2\kappa+1)(2\kappa+3)]/6(2\kappa-3)^{3}.\
$

We note that the term on the right hand side of the Eq. (7) is the
contribution of super-thermal electrons, positrons, and ions.

##  3 Derivation of NLSE

To study the MI of DAWs, we will derive the NLSE by employing the standard
multiple scale (reductive perturbation) technique [18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Let A be the state (column) vector
$(n_{d},u_{d},\phi)^{T}$, describing the system’s state at a given position x
and instant t. We shall consider small deviations from the equilibrium state
$A^{(0)}=(1,0,0)^{T}$ by taking [25, 26, 27]

$\displaystyle A=A^{(0)}+\epsilon
A^{(1)}+\epsilon^{2}A^{(2)}+\cdot\cdot\cdot=A^{(0)}+\sum_{n=1}^{\infty}\epsilon^{n}A^{(n)},\
$ (8)

where $\epsilon\ll 1$ is a smallness parameter. In the standard multiple scale
(reductive perturbation) technique, the stretched (slow) space and time
variables are commonly used by many authors [22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34] as follows:

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

where $v_{g}$ is the group velocity in the $x$ direction. We assume that all
perturbed states depend on the fast scales via the phase $\theta_{1}=kx-\omega
t$ only, while the slow scales enter the argument of the $l^{th}$ harmonic
amplitude $A_{l}^{(n)}$, which is allowed to vary along $x$,

$\displaystyle
A^{(n)}=\sum_{l=-\infty}^{\infty}A_{l}^{(n)}(\xi,\tau)e^{il(kx-\omega t)}.\ $
(10)

The reality condition $A_{-l}^{(n)}=A_{l}^{(n)*}$ is met by all state
variables. According to these considerations, the derivative operators are
treated as follows [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]

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

Now, by substituting (8)-(12) into (1), (2), and (7), and collecting the terms
containing $\epsilon$, the first order ($m=1$ with $l=1$) reduced equations
can be expressed as

$\displaystyle iku_{d1}^{(1)}=i\omega n_{d1}^{(1)},$ (13) $\displaystyle
ik\sigma_{1}n_{d1}^{(1)}=ik\phi_{1}^{(1)}+i\omega u_{d1}^{(1)},$ (14)
$\displaystyle n_{d1}^{(1)}=-k^{2}\phi_{1}^{(1)}-H_{1}\phi_{1}^{(1)},\ $ (15)

these equations reduce to

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

we thus obtain the dispersion relation for DAWs

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

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

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

with the compatibility condition

$\displaystyle
v_{g}=\frac{\omega^{2}-(\sigma_{1}k^{2}-\omega^{2})^{2}}{k\omega}.$ (21)

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

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

where

$\displaystyle
H_{4}=\frac{2H_{6}k^{2}(\sigma_{1}k^{2}-\omega^{2})^{2}-3\omega^{2}k^{4}-\sigma_{1}k^{6}}{2(\sigma_{1}k^{2}-\omega^{2})^{3}},$
$\displaystyle H_{5}=\frac{H_{4}\omega(\sigma_{1}k^{2}-\omega^{2})^{2}-\omega
k^{4}}{k(\sigma_{1}k^{2}-\omega^{2})^{2}},$ $\displaystyle
H_{6}=\frac{3\omega^{2}k^{4}+\sigma_{1}k^{6}-2H_{2}(\sigma_{1}k^{2}-\omega^{2})^{3}}{6k^{2}(\sigma_{1}k^{2}-\omega^{2})^{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

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

where

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

Finally, the third harmonic modes ($m=3$) and ($l=1$), with the help of
(16)-(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\mid\Phi\mid^{2}\Phi=0,\
$ (28)

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

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

The nonlinear coefficient $Q$ is

$\displaystyle
Q=\frac{3H_{3}(\sigma_{1}k^{2}-\omega^{2})^{2}+2H_{2}(H_{6}+H_{9})(\sigma_{1}k^{2}-\omega^{2})^{2}-2\omega
k^{3}(H_{5}+H_{8})-(\sigma_{1}k^{4}+k^{2}\omega^{2})(H_{4}+H_{7})}{2\omega
k^{2}}.\ $

It is important to mention few more points: The reductive perturbation
technique can also be used to derive the Korteweg-de Vries equation for
describing the evolution of a non-modulated waves, i.e. a bare pulse with no
fast oscillations inside the packet. However, the well known nonlinear
mechanism involved in plasma wave dynamics is amplitude modulation, which may
be due to parametric wave coupling, interaction between high and low frequency
modes or simply to the nonlinear self-interaction of the carrier waves. The
standard method to study this mechanism adopts a multiple scales perturbation
(also known as reductive perturbation [18, 19]) technique, which generally
leads to a NLSE describing the evolution of a slowly varying wave packet or
envelope. The wave packet may undergo a Benjamine-Feir type MI under certain
conditions. The MI of wave packets in plasmas acts as a precursor for the
formation of bright and dark envelope solitons.

Figure 1: Plot of $P/Q$ vs $k$ for different values of $\sigma_{1}$, when
$\sigma_{2}=0.2$, $\sigma_{3}=1.3$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$, and
$\kappa=1.8$.

Figure 2: Plot of $P/Q$ vs $k$ for different values of $\sigma_{2}$, when
$\sigma_{1}=0.007$, $\sigma_{3}=1.3$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$, and
$\kappa=1.8$.

Figure 3: Plot of $P/Q$ vs $k$ for different values of $\sigma_{4}$, when
$\sigma_{1}=0.007$, $\sigma_{2}=0.2$, $\sigma_{3}=1.3$, $\sigma_{5}=0.6$, and
$\kappa=1.8$.

Figure 4: Plot of $P/Q$ vs $k$ for different values of $\kappa$, when
$\sigma_{1}=0.007$, $\sigma_{2}=0.2$, $\sigma_{3}=1.3$, $\sigma_{4}=0.7$, and
$\sigma_{5}=0.6$.

##  4 Modulational instability

The stable and unstable domains of the DAWs are organized by the sign of the
dispersion ($P$) and nonlinear ($Q$) coefficients of the standard NLSE (28).
The stability of DAWs in four components DPM is governed by the sign of $P$
and $Q$ [27, 35, 39, 40, 41]. When $P$ and $Q$ are same sign (i.e., $P/Q>0$),
the evolution of the DAWs amplitude is modulationally unstable. On the other
hand, when $P$ and $Q$ are 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 domains for the DAWs. The point,
at which transition of $P/Q$ curve intersect with $k$-axis, is known as
threshold or critical wave number $k$ ($=k_{c}$).

The effects of the plasma parameters, specially, the charge state and
temperature of the plasma species (via $\sigma_{1}$) can be observed in Fig.
2, and it is obvious that (a) the unstable window, at which MI sets for the
DAWs and allows the formation of the bright envelope solitons, for the DAWs
opens for the large values of $k$ ($k>k_{c}$); (b) the stable domain for the
DAWs can be found for small values of $k$ ($k<k_{c}$), and allows the
formation of the dark envelope solitons; (c) when $\sigma_{1}=0.005$, $0.007$,
and $0.009$ then the corresponding $k_{c}$ value is $k_{c}\equiv 2.45$ (dotted
blue curve), $k_{c}\equiv 2.3$ (dashed green curve), and $k_{c}\equiv 2.2$
(solid red curve); (d) so, the $\sigma_{1}$ reduces the stable domain of the
DAWs; (e) actually, dust (ion) temperature reduces (increases) the stable
domain of the DAWs for the constant value of dust charge state (via
$\sigma_{1}=3T_{d}/Z_{d}T_{i}$).

Figure 2 indicates the effects of the number density of the positrons and dust
grains in recognizing the stable and unstable domains of the DAWs. It is clear
from this figure that (a) the stable domain enhances with the increase in the
value of positron number density whereas negatively charged dust number
density suppresses the stable domain for the DAWs when the charge state of the
negative dust remains invariant (via $\sigma_{2}=n_{p0}/Z_{d}n_{d0}$); (b) the
unstable domain enhances with $Z_{d}$ for constant value of $n_{p0}$ and
$n_{d0}$. So, the number density and the charge state of the negative dust
grains rigourously can change the stable and unstable domains of the DAWs.

The stability domains of the DAWs in four components DPM is also organized by
the temperature of the plasma parameters. Figure 4 describes that the stable
(unstable) domain of the DAWs increases with an increase in the value of the
ion (electron) temperature (via $\sigma_{4}=T_{i}/T_{e}$). The instability
criterion of the DAWs for super-thermality of electrons, positrons, and ions
in four components EPID plasma can be observed in Fig. 4, and it can be seen
from this figure that (a) when for $\kappa=1.8$, $2.0$, and $2.2$ the
corresponding $k_{c}$ value is $k_{c}\equiv 2.6$ (dotted blue curve),
$k_{c}\equiv 2.3$ (dashed green curve), and $k_{c}\equiv 2.2$ (solid red
curve); (b) so, the $k_{c}$ value decreases with the increase of $\kappa$ and
this result is a good agreement with the result of Ahmed et al. [11] work.

##  5 Envelope solitons

The bright (when $P/Q>0$) and dark (when $P/Q<0$) envelope solitonic
solutions, respectively, can be written as [27, 35, 36, 37, 38, 39, 40, 41]

$\displaystyle\Phi(\xi,\tau)=\left[\psi_{0}~{}\mbox{sech}^{2}\left(\frac{\xi-U\tau}{W}\right)\right]^{\frac{1}{2}}\times\exp\left[\frac{i}{2P}\left\\{U\xi+\left(\Omega_{0}-\frac{U^{2}}{2}\right)\tau\right\\}\right],$
(29)
$\displaystyle\Phi(\xi,\tau)=\left[\psi_{0}~{}\mbox{tanh}^{2}\left(\frac{\xi-U\tau}{W}\right)\right]^{\frac{1}{2}}\times\exp\left[\frac{i}{2P}\left\\{U\xi-\left(\frac{U^{2}}{2}-2PQ\psi_{0}\right)\tau\right\\}\right],$
(30)

Figure 5: Plot of $Re(\Phi)$ vs $\xi$ when $\sigma_{1}=0.007$,
$\sigma_{2}=0.2$, $\sigma_{3}=1.3$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$,
$\kappa=1.8$, $k=2.5$, $\tau=0$, $U=0.4$, $\Omega_{0}=0.4$, and
$\psi_{0}=0.005$.

Figure 6: Plot of $Re(\Phi)$ vs $\xi$ when $\sigma_{1}=0.007$,
$\sigma_{2}=0.2$, $\sigma_{3}=1.3$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$,
$\kappa=1.8$, $k=1.5$, $\tau=0$, $U=0.4$, $\Omega_{0}=0.4$, and
$\psi_{0}=0.005$.

Figure 7: Plot of $|\Phi|$ vs $\xi$ for different values of $\sigma_{3}$, when
$\sigma_{1}=0.007$, $\sigma_{2}=0.2$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$,
$\kappa=1.8$, $k=2.5$, $\tau=0$, $U=0.4$, $\Omega_{0}=0.4$, and
$\psi_{0}=0.005$.

Figure 8: Plot of $|\Phi|$ vs $\xi$ for different values of $\sigma_{3}$, when
$\sigma_{1}=0.007$, $\sigma_{2}=0.2$, $\sigma_{4}=0.7$, $\sigma_{5}=0.6$,
$\kappa=1.8$, $k=1.5$, $\tau=0$, $U=0.4$, $\Omega_{0}=0.4$, and
$\psi_{0}=0.005$.

where $\psi_{0}$ indicates the envelope amplitude, $U$ is the traveling speed
of the localized pulse, $W$ is the pulse width which can be written as
$W[=(2P\psi_{0}/Q)^{1/2}]$, and $\Omega_{0}$ is the oscillating frequency for
$U=0$. We have depicted bright envelope solitons in Figs. 6 and 8 by
numerically analyzed of equation (29). We have also numerically analyzed (30)
in Figs. 6 and 8. Figure 6 indicates that (a) the MI of the DAWs associated
with unstable domain (i.e., $P/Q>0$) leads to generate a bare pulse with fast
oscillations inside the packet; (b) initially, the hight of the bare pulse is
maximum at $\xi=0$ but decreases with the increase in the value of positive
and negative $\xi$, and still tends to zero for large value of $\xi$ (i.e.,
$\xi\geq\pm 10$). On the other hand, Fig. 6 describes that (a) the MI of the
DAWs associated with stable domain (i.e., $P/Q<0$) leads to generate a bare
pulse with fast oscillations inside the packet; (b) initially, the hight of
the bare pulse is minimum at $\xi=0$ but increases with the increase in the
value of positive and negative $\xi$, and finally, remains constant for large
value of $\xi$ (i.e., $\xi\geq\pm 2$).

The nonlinear and dispersion property of the plasma medium as well as the
mechanism of the formation of bright envelope solitons are totally depended on
the various plasma parameters. It is obvious from Fig. 8 that (a) the width of
the bright envelope solitons decreases with the increase in the value of
positively charged ion number density (i.e., $n_{i0}$) while increases with
the increase in the value of negatively charged dust grains number density
(i.e., $n_{d0}$) when their charge state (i.e., $Z_{i}$ and $Z_{d}$) remain
constant (viz., $\sigma_{3}=Z_{i}n_{i0}/Z_{d}n_{d0}$); (b) the magnitude of
the amplitude remain invariant. So, the amplitude is not affected by the
variation of the $\sigma_{3}$.

The influence of the number density of the positively charged ions and
negatively charged massive dust grains as well as their charge state on the
formation of dark envelope solitons can be observed in Fig. 8, and it is
obvious from this figure that the increase in the value of $\sigma_{3}$ causes
to change the width of the dark envelope solitons but does not cause any
change in the magnitude of the amplitude of the dark envelope solitons (via
$\sigma_{3}=Z_{i}n_{i0}/Z_{d}n_{d0}$). So, the plasma parameters, viz.,
$n_{i0}$, $n_{d0}$, $Z_{i}$, $Z_{d}$, and $k$ play a vital role in recognising
the structure of the dark envelope solitons.

##  6 Conclusion

We have investigated the characteristics of the amplitude modulation of DAWs
by using a NLSE, which is successfully derived by employing the standard
reductive perturbation technique in a four components DPM composed of
$\kappa$-distributed electrons, positrons, ions, and negatively charged
adiabatic dust grains. In the formation and propagation of DAWs, the moment of
inertia is provided by the mass of the adiabatic warm dust grains and
restoring force is provided by the thermal pressure of the super-thermal
electrons, positrons, and ions. So, each of the plasma components of the DPM
provides a large contribution in the formation and propagation of the DAWs in
four components DPM. A number of authors [1, 2, 3] have studied DASWs in a
four components DPM having electrons, positrons, ions, and negatively charged
dust grains. However, they have not studied the nonlinear behaviour of DA wave
packets and their MI. Therefore, we have considered this plasma system to
investigate the MI of DA wave packet and the formation of the bright and dark
envelope solitons in the modulatonally unstable and stable domains of DAWs,
respectively. We have also observed that the amplitude of the DA envelope
solitons remains constant but the width of the DA envelope solitons changes
with the variation of the various plasma parameters. Our results has a good
agreement with the previous work [10].

To conclude, we hope that our results may be helpful for understanding the MI
of DAWs and associated the bright and dark envelope solitons in space plasmas,
viz., pulsar magnetosphere, supernova environments, galactic nuclei and also
in the laboratory plasmas, viz., intense laser beams. It may be noted here
that the gravitational and magnetic effects are very important to consider but
beyond the scope of our present work. In future and for better understanding,
someone can investigate the propagation of nonlinear waves in a four
components DPM by considering the gravitational and magnetic effects.

## acknowledgements

The authors are grateful to the anonymous reviewer for his/her constructive
suggestions which have significantly improved the quality of our manuscript.
A. A. Noman is thankful to the Bangladesh Ministry of Science and Technology
for awarding the National Science and Technology (NST) Fellowship. A. Mannan
thanks the Alexander von Humboldt Foundation for a Postdoctoral Fellowship.

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