Dust-acoustic envelope solitons in super-thermal plasmas

TL;DR

This paper investigates how super-thermal particles influence the stability of dust-acoustic waves in a complex plasma, revealing how their presence alters the conditions for modulational instability through nonlinear analysis.

Contribution

It introduces a detailed analysis of dust-acoustic wave stability in super-thermal plasmas using the nonlinear Schrödinger equation and identifies how super-thermal particles modify stability regions.

Findings

Super-thermal particles significantly alter stable and unstable wave regions.

The critical wave number for instability depends on particle concentrations.

Two distinct parametric regions for wave stability are identified.

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 Schr\"{o}dinger 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.

Equations

$$\frac{\partial n_d}{\partial t}+\frac{\partial }{\partial x}(n_d{u}_d)=0,$$

$$\frac{\partial u_d}{\partial t}+u_d\frac{\partial u_d}{\partial x}+{\sigma }_1{n}_d\frac{\partial n_d}{\partial x}=\frac{\partial \varphi }{\partial x},$$

$$\frac{{\partial }^2\varphi }{\partial x^2}=({\sigma }_2+{\sigma }_3-1)n_e-{\sigma }_2{n}_p+n_d-{\sigma }_3{n}_i,$$

$$n_e={\left[1-\frac{{\sigma }_4\varphi }{({\kappa }_e-3/2)}\right]}^{-\kappa +\frac{1}{2}},$$

$$n_p={\left[1+\frac{{\sigma }_5\varphi }{({\kappa }_p-3/2)}\right]}^{-\kappa +\frac{1}{2}},$$

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

$$\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,\$$

$$H_1=({\sigma }_2+{\sigma }_3-1){\gamma }_1{\sigma }_4+{\gamma }_1{\sigma }_2{\sigma }_5+{\gamma }_1{\sigma }_3,H_2=({\sigma }_2+{\sigma }_3-1){\gamma }_2{\sigma }_4^2-{\gamma }_2{\sigma }_2{\sigma }_5^2-{\gamma }_2{\sigma }_3,$$

$$H_3=({\sigma }_2+{\sigma }_3-1){\gamma }_3{\sigma }_4^3+{\gamma }_3{\sigma }_2{\sigma }_5^3+{\gamma }_3{\sigma }_3,{\gamma }_1=(2\kappa -1)/(2\kappa -3),$$

$$\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}.\$$

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

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

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

$$\frac{\partial }{\partial t}\to \frac{\partial }{\partial t}-ϵv_g\frac{\partial }{\partial \xi }+{ϵ}^2\frac{\partial }{\partial \tau },$$

$$\displaystyle\frac{\partial}{\partial x}\rightarrow\frac{\partial}{\partial x}+\epsilon\frac{\partial}{\partial\xi}\,.\$$

$$ik{u}_{d1}^{(1)}=i\omega n_{d1}^{(1)},$$

$$ik{\sigma }_1{n}_{d1}^{(1)}=ik{\varphi }_1^{(1)}+i\omega u_{d1}^{(1)},$$

$$\displaystyle n_{d1}^{(1)}=-k^{2}\phi_{1}^{(1)}-H_{1}\phi_{1}^{(1)},\$$

$$n_{d1}^{(1)}=\frac{k^2}{{\sigma }_1{k}^2-{\omega }^2}{\varphi }_1^{(1)},$$

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

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

$$n_{d1}^{(2)}=\frac{k^2}{{\sigma }_1{k}^2-{\omega }^2}{\varphi }_1^{(2)}-\frac{2ik\omega (k{v}_g-\omega )}{({\sigma }_1{k}^2-{\omega }^2{)}^2}\frac{\partial {\varphi }_1^{(1)}}{\partial \xi },$$

$$\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},\$$

$$v_g=\frac{{\omega }^2-({\sigma }_1{k}^2-{\omega }^2{)}^2}{k\omega }.$$

$$n_{d2}^{(2)}=H_4|{\varphi }_1^{(1)}{|}^2,$$

$$u_{d2}^{(2)}=H_5|{\varphi }_1^{(1)}{|}^2,$$

$$\displaystyle\phi_{2}^{(2)}=H_{6}|\phi_{1}^{(1)}|^{2},\$$

$$H_4=\frac{2H_6{k}^2({\sigma }_1{k}^2-{\omega }^2{)}^2-3{\omega }^2{k}^4-{\sigma }_1{k}^6}{2({\sigma }_1{k}^2-{\omega }^2{)}^3},$$

$$H_5=\frac{H_4\omega ({\sigma }_1{k}^2-{\omega }^2{)}^2-\omega k^4}{k({\sigma }_1{k}^2-{\omega }^2{)}^2},$$

$$\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}}.\$$

$$n_{d0}^{(2)}=H_7|{\varphi }_1^{(1)}{|}^2,$$

$$u_{d0}^{(2)}=H_8|{\varphi }_1^{(1)}{|}^2,$$

$$\displaystyle\phi_{0}^{(2)}=H_{9}|\phi_{1}^{(1)}|^{2},\$$

$$H_7=\frac{H_9({\sigma }_1{k}^2-{\omega }^2{)}^2-2v_g\omega k^3-{\sigma }_1{k}^4-k^2{\omega }^2}{({\sigma }_1{k}^2-{\omega }^2{)}^2({\sigma }_1-v_g^2)},$$

$$H_8=\frac{v_g{H}_7({\sigma }_1{k}^2-{\omega }^2{)}^2-2\omega k^3}{({\sigma }_1{k}^2-{\omega }^2{)}^2},$$

$$\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})}.\$$

$$\displaystyle i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q\mid\Phi\mid^{2}\Phi=0,\$$

$$\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})}.\$$

$$\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}}.\$$

$$\Phi (\xi ,\tau )={\left[{\psi }_0\text{mbox}{sech}^2\left(\frac{\xi -U\tau }{W}\right)\right]}^{\frac{1}{2}}\times \exp \left[\frac{i}{2P}\left\{U\xi +\left({\Omega }_0-\frac{U^2}{2}\right)\tau \right\}\right],$$

Full text

††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:

[TABLE]

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

[TABLE]

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

[TABLE]

where

[TABLE]

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]

[TABLE]

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:

[TABLE]

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 $x$,

[TABLE]

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]

[TABLE]

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

[TABLE]

these equations reduce to

[TABLE]

we thus obtain the dispersion relation for DAWs

[TABLE]

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

[TABLE]

with the compatibility condition

[TABLE]

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

[TABLE]

where

[TABLE]

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

[TABLE]

where

[TABLE]

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:

[TABLE]

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

[TABLE]

The nonlinear coefficient $Q$ is

[TABLE]

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.

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]

[TABLE]

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.