Various Aspects of Dust-Acoustic Solitary Waves (DAWs) in Inhomogeneous Plasmas

Abstract

Dusty plasma offers an extensive study of space and astrophysical environments. In this chapter, we have studied some of the basic properties of dusty plasmas, interaction of dust and plasma particles, and the effect of intramolecular attraction and repulsion between these plasma and dust grain particles. During these investigations, we have discussed a few basic relations of dusty plasma and the effect of dust particles on the nonlinear wave structures in dusty plasma. Here, we have also studied the various aspects of dust-acoustic solitary waves (DASWs) in inhomogeneous plasma. To study and analyze the various aspects of DAWs in inhomogeneous plasmas, the governing fluid equations of plasmas are considered to derive the Korteweg de-Vries (KdV) equation. The solution of the KdV equation is obtained as soliton or solitary wave. The solitary wave solution indicates the various characteristics of DASWs in the inhomogeneous dusty plasma. In this chapter, a systematic and extensive study on DAWs is also included for the inhomogeneous and unmagnetized plasmas.

1. Introduction

Plasmas and dust particles are the two fundamental elements of the universe. The interaction of these two basic ingredients forms a new area of the research field called dusty plasma or complex plasma. Dusty plasma is a complex plasma system consisting of plasma and charged dust particles. Dusty plasmas are abundant everywhere in the universe. Many examples of dusty plasma can be seen in the visible universe—Saturn rings, interstellar clouds and circumstellar clouds, comets, and Mars rings [1]. In the present days of plasma research, dusty plasma is also one of the advanced and trending research fields. As an application of dusty plasma, it includes the study of geophysics, space missions, advancement in astrophysical areas such as a better understanding of the formation of galaxies and stars or planets, and applications in the industries for producing modern materials. Besides the astrophysical environments, dusty plasma is also considered one of the essential fields to study and work in controlled thermonuclear research (CTR). In 1982, the Voyager spacecraft [2] discovered some interesting facts about Saturn’s B ring and the radial spokes for the first time. Including the astrophysical issues, dusty plasma has excellent applications in manufacturing new materials in semiconductor industries [3].

Many researchers have done both theoretical and experimental works in dusty plasma. To understand the fundamental properties of duty plasma, some devices, such as the rotating drum [4] and dust shaker systems [5], are newly introduced in the laboratory. Researchers like Northrop [6] and Goertz [7] have added some theoretical works in the field of the dusty plasma system. In unmagnetized dusty plasmas, usually, there are three kinds of normal modes of waves considered. They are dust-acoustic waves (DAW) [8], dust ion-acoustic waves (DIAW) [9], and dust-lattice waves (DLW) [10]. Some theoretical and empirical studies [11, 12] have been done to understand the linear behaviors of dusty plasma in various physical situations. Some of the researchers have done their work on the instabilities of nonlinear waves in dusty plasmas [13, 14, 15, 16, 17, 18, 19, 20]. For studying the linear wave theories for small amplitude waves in plasma physics, the nonlinearities are omitted, but the nonlinearities cannot be omitted for the high-amplitude waves. The existence of nonlinearity indicates some novel behaviors of the nonlinear wave structures, such as solitons, supersolitons, rouge waves, and shock waves. Theoretically, the various properties of dust lattice solitary waves [10, 21], DA solitary waves [9, 22, 23] DIA solitary waves [24, 25], and shock waves [26, 27, 28, 29] have been studied in an inhomogeneous plasma. Complex plasmas have extensive applications in astrophysical systems, space science, and laboratory experiments [30, 31]. Atteya et al. studied the properties of DASWs consisting of negatively and positively charged dust grains in the presence of ᴋ - distributed superthermal electrons [32]. Earlier, many research works have been done based on homogeneous plasma systems. Still, in the present days, inhomogeneous plasma systems are also widely discussed to understand the various applications of dusty plasma in space and laboratory discharges [33]. The dust-cyclotron wave and modulational instability of dust-acoustic solitary waves have been investigated in the three-component plasma for electrons, ions, and dust particles in a magnetic field [34]. Recently, a study has been done on magneto-acoustic waves propagating and pair-ion fullerene plasma in linear and nonlinear environments [35]. The dust grains are negatively charged and unstable due to their concentration of weights [36]. With the increase in the electron density, the phase velocity of the dust ion-acoustic solitary waves decreases [37, 38]. The DIASWs were also studied both theoretically and experimentally. The DIASW was studied in the presence of a collisionless plasma [39] for some limiting cases, but it was not well designed in some real conditions [40]. Ghosh et al. [41] investigated the effect of damping on DIAWs. Losseva et al. [42] studied the evolution of weakly dissipative hybrid DIASW in the presence of collisional plasma. Popel et al. [43] experimentally investigated the wave velocity, which is much larger than the wave velocity in theoretical conditions. Kruskal [44] has also observed the gravitational wave for inviscid fluid that can asymptotically describe the Korteweg-de Vries (KdV) equation. Some researchers have studied solitary waves, DIASWs, and shock waves both theoretically and experimentally [11, 45, 46, 47] in dusty plasmas. Ashraf et al. [48] observed the characteristics of obliquely propagating DIASWs in the presence of nonextensive dusty plasma. Bacha et al. [49] studied the DIASW in the presence of non-Maxwellian dusty plasmas. Tribeche and Zerguini [50] studied the effect of modulations and instabilities in DIASWs in the presence of collisional dusty plasma. Later, an investigation revealed that a plasma system consisting of Gaussian-distributed charged dust grains exhibits DIASWs properties in the presence of a collisional effect [51].

Similarly, a few works have been established [52, 53, 54, 55] on the solution of the damped KdV equation in various plasma systems using the classical fluid approach of plasma physics. The application of plasma particles is observed under the effect of nonextensive electron distributions in astrophysical and cosmological scenarios [56], plasma dynamics [57], nonlinear gravitational model [58], and Hamiltonian systems [59] in the presence of long-range interaction. The impact of nonextensive plasma particles on various waves propagating in inhomogeneous plasma is studied in works [60, 61, 62, 63]. An extensive study on the distribution of nonextensive electrons is seen in the results [64].

This chapter discusses some essential characteristics of dusty plasma and the various aspects of the DASWs in the inhomogeneous plasma system. We have presented our study and analysis of the DASWs in inhomogeneous plasmas. During the investigations, we considered the governing fluid equations to derive the Korteweg-de Vries (KdV) equation. The characteristic behaviors of DASWs and the interaction of dust and plasma particles are also studied in various physical situations.

2. Properties of dusty plasmas

The dust particles consist of some fundamental elements, such as electrons, ions, and charged dust particles. In a dusty plasma system, there are three characteristic scale lengths of the complex plasma system. They are the average intergrain distance, the Debye radius, and the dust grain radius. In a dusty plasma system, the Debye radius ${\lambda }_D$ is given by [65]

$$\frac{1}{{{\lambda }^2}_D}=\frac{1}{{{\lambda }^2}_{\mathit{De}}}+\frac{1}{{{\lambda }^2}_{\mathit{Di}}}$$E1

where ${\lambda }_{\mathit{Di}}$ and ${\lambda }_{\mathit{De}}$ are the ion Debye radius and electron Debye radius of a dusty plasma system, respectively. In a dusty plasma mixture, the dust grain particles are one of the most fundamental and essential elements. These dust grain particles are comparatively larger than the negatively charged electrons or positively charged ions in the plasma medium. In a dusty plasma system, the quasi-neutrality condition for the negatively charged dust grains is given by

$$n_{i0}=n_{e0}+Z_{d0}{n}_{d0}$$E2

where $Z_{d0}$ is the unperturbed number density of charged dust grains and $n_{k0}$ is the unperturbed number density of the particle species k. Here, the particle species k stands for species of electrons, ions, and dust gains, respectively (I = ions, e = electrons, and d = dust grains). When the dust grains and electrons are attached, we have the following equation:

$$n_{e0}\ll Z_{d0}{n}_{d0}$$E3

However, due to the presence of a minimum value ratio of the number densities of ions to the electrons, the reduction of electrons cannot be done completely. While the surface potential of the dust particles tends to zero, the ratio of the number densities of the ions and electrons is not seen to have vanished. Usually, these types of situations, that is, depletion of electrons occur in the Saturn rings, as well as in the laboratory discharges. However, in the thermal complex plasmas, the dust particles get positively charged by releasing electrons. So, by omitting the electrons from charged dust particles, the dust particles become positively charged. Therefore, at an equilibrium condition, we get the following equation:

$$n_{e0}\approx Z_{d0}{n}_{d0}$$E4

The Coulomb coupling parameter for the strongly correlated dust particles in a dusty medium is given by

$$\mathrm{\Gamma }=\frac{q^2}{d{T}_d}exp\left(-\kappa \right)$$E5

where, $q=Z_{d0}e$ is the unperturbed number density of dust particles, $\kappa =\frac{d}{{\lambda }_D}$,and $T_d$ is the temperature of dust grains. In the case of weakly coupled dusty plasma, we have $\mathrm{\Gamma }\ll 1$, and for strongly coupled dusty plasma, it is found to be $\kappa \ll 1$ and $\mathrm{\Gamma }\gg 1$. To understand the dynamics and crystal formations in dusty plasma, the strongly coupled dusty plasmas are kept along in the low-temperature laboratory discharges in strongly coupled dusty plasmas. In strongly coupled dusty plasmas, the colloidal systems and laser implosion experiments both show a strong linkage between them.

3. Interactions of dust and plasma particles

The size of the dust particles or dust grains differs from the other plasma particles, that is, from ions to electrons. The dust particles are very large compared to these plasma particles. They are dependent on their surroundings. The size of the dust charges depends on the inclusion of ions, electron fluxes, and the local parameters, but the plasma particles are depending on the distribution functions. The fluxes are influenced by the inclusion of dust particles, so there is also a change in the charges of dust grains. Thus, the changes in the interaction of dust particles with other plasma particles are seen accordingly.

The existence of new forces between the plasma particles and dust grains can also be experienced even if there are fixed charges. In this case, the interaction between the dust grain particles and plasma particles could be different. The interactions among the dust particles could vary due to the variations in the distance of the dust particles and Debye length. These can be experimented by Etching experiments and plasma-dust crystal experiments. The distance between the dust particles and the Debye length is very less in the Etching experiment, but on the other hand, the distance between the dust particles and the Debye length is very large in plasma-dust crystal experiment. In plasma-dust crystal experiments, the Coulomb interaction and mutual shadowing of plasma fluxes both are given much importance.

4. Effect of intramolecular dust particle attraction and repulsion

The nonlinear structures of dusty plasma can redistribute the dust particles in their structures. We may assume that the dust particles will be concentrated in a single place in the nonlinear structure. Thus, it will result in the dust particle as negatively charged dust grains due to the fluctuations of electrons and ions. In the case of dusty plasma, there is no interaction of dust particles with neutral particles. Then the structures of the nonlinear DA waves will be modified by the dust particles that are to be drift vortices. If there is an interaction between the dust particles and neutral gas particles, there will be a strong bond of vortices. It will exhibit the simultaneous drift and gas vortices of the nonlinear wave structures.

In the upper atmosphere and the lower ionosphere, there may exist some nonlinear wave structures due to the existence of gas and drift vortices. Due to the existence of a degree of pollution and dust particles in the upper atmosphere, some motion can be observed in the linkage to the usual vortex motion. The solar activity and degree of ionization influence the drift structures of the nonlinear waves in the inhomogeneous duty plasma systems.

5. Basic relations in dusty plasma

1. Debye length $\left({\lambda }_D\right)$ and dust size $\left(D\right):$ In the case of smaller dust particles, that is, $D\ll {\lambda }_D$, we have obtained the Eq. (1) as follows:

   $$\frac{1}{{{\lambda }^2}_D}=\frac{1}{{{\lambda }^2}_{\mathit{De}}}+\frac{1}{{{\lambda }^2}_{\mathit{Di}}}.$$E6

   Then for $D={\lambda }_{\mathit{Di}}$ and $T_e\gg T_i$, we have obtained the dimensionless equation for dust size $D$ as follows:

   $$D=\frac{D_0}{{\lambda }_{\mathit{Di}}}$$E7

2. Charging of a dust particle: Floating potential argument shows an assumption on the charge of the dust particle in the units of electron charge $e$ is given by $\frac{Z_d{e}^2}{D}\sim T_e$. Then the dimensionless equation for the dust charge $z$ is given by

   $$z\equiv \frac{Z_d{e}^2}{D{T}_e}$$E8

3. Dimensionless dust density: In the dust-plasma crystal experiment, the dimensionless dust density $\rho$ is of order 1; in the etching experiment, it is of order 10; and in the astrophysical situations, it varies from $10$ to ${10}^{-3}$, which is shown by the equation given by

$$\rho \equiv \frac{n_d{Z}_d}{n_e}$$E9

where $n_d$ is the normalized number density of dust particles. The quasi-neutrality condition for dusty plasma is given by

$$n_{i,0}=n_{e,0}\left(1+\rho \right)$$E10

6. Methodology to solve nonlinear equations

Nonlinear plasma waves follow governing fluid equations of plasma in various physical circumstances. Due to the existence of nonlinear plasma inhomogeneity, plasma is considered to be one of the frontier research areas in science and engineering. Using fluid approach in plasma physics, lots of research has been carried out for understanding the soliton propagation, soliton reflection, sheath and beam formation, etc. For the last many decades, researchers are using one of the major techniques (method) for deriving and solving nonlinear waves in plasma physics, which is reductive perturbation technique (RPT). This RPT is very useful for the nonlinear small amplitude waves. In this technique, both the space and time coordinates are transformed to new stretched coordinates for obtaining the family of nonlinear equations, such as Korteweg-de Vries (KdV) equation, nonlinear Schrödinger equation (NLSE), and Kadomtsev Petviashvili (KP) equation.

For dust-acoustic waves (DAWs), the governing fluid equations are considered as follows:

Continuity equation:

$$\frac{\partial n_d}{\partial t}+\frac{\partial \left(n_d{v}_d\right)}{\partial x}=0$$E11

Momentum equation:

$$\frac{\partial v_d}{\partial t}+n_d\frac{\partial v_d}{\partial x}+q\frac{\partial \varnothing }{\partial x}=0$$E12

Electron distribution equation:

$$n_e=exp\left(\varphi \right)$$E13

Ion distribution equation:

$$n_i=exp\left(-\varphi \right)$$E14

Poison equation:

$$\frac{{\partial }^2\varphi }{{\partial }^{2}x}=n_e-n_i+q{n}_d$$E15

In this technique, we have considered a set of dependent variables in terms of power of smallness parameter $\mathit{\epsilon }$. Then using the set of stretched coordinates [66] $\xi ={\epsilon }^{1/2}\left(x-\mathit{\lambda t}\right),\tau ={\epsilon }^{3/2}t$, we get the k-dv equation as follows:

$$\frac{\partial {\varphi }_1}{\partial \tau }+A{\varphi }_1\frac{\partial {\varphi }_1}{\partial \xi }+B\frac{{\partial }^3{\varphi }_1}{\partial {\xi }^3}=0$$E16

In the above K-dV equation, the constants $\mathit{A}$ and $\mathit{B}$ represent the nonlinearity and dispersive effect, respectively, in a considered plasma system. After solving the above equation and using some particular transformations, we get the required soliton solution ${\mathit{\varphi }}_{\mathbf{1}}.$

7. Effects of dust particles on the nonlinear wave structures

Dust acoustic (DA) solitary waves: The dimensionless nonlinear governing fluid equations for the DA waves in inhomogeneous unmagnetized plasmas are given by [9]

$$\frac{\partial n_d}{\partial t}+\frac{\partial \left(n_d{u}_d\right)}{\partial x}=0$$E17

$$\frac{\partial u_d}{\partial t}+u_d\frac{\partial u_d}{\partial x}=-\frac{\partial \varnothing }{\partial x}$$E18

$$\frac{{\partial }^2\varphi }{\partial x^2}=n_d+{\gamma }_e{\mathrm{e}}^{{\mathrm{\sigma }}_{\mathrm{i}}\mathrm{\varphi }}-{\gamma }_i{\mathrm{e}}^{-\mathrm{\varphi }}$$E19

where $u_d$ is the normalized fluid velocity of dust grains, $\gamma =\raisebox{1ex}{${\mathrm{n}}_{\mathrm{e}0}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{n}}_{\mathrm{i}0}$}\right.$, and $\varphi$ is the normalized electrostatic wave potential$.$ Here, $x$ and $t$ are the space variable and time variables normalized by ${\lambda }_D$and ion plasma period $\left(\omega \right)$, respectively. We have considered that the electrons and ions are distributed in the Boltzmannean distribution functions in accordance. We have also considered ${\gamma }_e=\gamma /\left(1-\gamma \right)$, ${\mathrm{\sigma }}_{\mathrm{i}}=T_i/T_e,$ and ${\gamma }_i=1/\left(1-\gamma \right)$.

To employ the reductive perturbation technique (RPT), the stretched coordinates [66] are chosen as $\xi ={ϵ}^{\frac{1}{2}}\left(z-\mathit{Mt}\right)$ and $\tau ={ϵ}^{3/2}t$, where $ϵ$ is a smallness expansion parameter and M is the normalized phase velocity of the soliton. The expanded dependent variables $n_d,v_d,Z_d,$and $\varphi$ in terms of power series of $ϵ$ are given by:

$$\left[\begin{array}{c}{n}_d\\ v_d\\ \varphi \\ Z_d\end{array}\right]=\left[\begin{array}{c}{n}_{d0}\\ 0\\ 0\\ Z_{d0}\end{array}\right]+ϵ\left[\begin{array}{c}{n}_{d1}\\ v_{d1}\\ {\varphi }_1\\ Z_{d1}\end{array}\right]+{ϵ}^2\left[\begin{array}{c}{n}_{d2}\\ v_{d2}\\ {\varphi }_2\\ Z_{d2}\end{array}\right]+\dots$$E20

Now, using (20) and RPT on the Eqs. (17)–(19), we have obtained the Korteweg-de Vries (KdV) equation, which is of the form

$$\frac{\partial {\varphi }^{\left(1\right)}}{\partial \tau }+U\frac{\partial {\varphi }^{\left(1\right)}}{\partial \xi }+V\frac{{\partial }^3{\varphi }^{\left(1\right)}}{\partial {\xi }^3}=0$$E21

where $U=-\frac{M^3}{{\left(1-\gamma \right)}^2}\left[1+\left(3+{\mathrm{\sigma }}_{\mathrm{i}}\gamma \right)\gamma {\mathrm{\sigma }}_{\mathrm{i}}+0.5\gamma \left(1+{{\mathrm{\sigma }}_{\mathrm{i}}}^2\right)\right]$ and $V=0.5M^3.$

8. Results and discussions

Solving the Eq. (21), we get the solitary wave solution or soliton w.r.t. The velocity $v$ is given by [37]

$${\varphi }^{\left(1\right)}={\varphi }_m\mathit{sec}{h}^2\left[\frac{U}{W}\right]$$E22

where $P=\xi -v\mathrm{\tau }$, $A=0.5{\gamma }^{-1.5}$, and $v$ is the normalized constant velocity, ${\varphi }_m=\frac{3v}{c}$ and $W=\sqrt{\frac{4A}{v}}$ are amplitude and width of the DASWs, respectively. As $M>0$ and Eq. (22) indicates and we always get the result $c<0$ for every possible value of ${\mathrm{\sigma }}_{\mathrm{i}}$ and $\gamma$. Thus, Eq. (22) also indicates that the dust acoustic (DA) solitary waves are also included in dusty plasma for the negative potential.

From the above Figure 1, we have observed that the amplitude of dust-acoustic (DA)

Figure 1.

Variations in amplitude of DASW w.r.t. number density of dust grains ($n_{d0}$).

solitary wave decreases with the increase in the number of dust grains. This implies that if the density of dust particles is compact in a particular region of space in the plasma system, the soliton will propagate smoothly with a lower amplitude wave. In this case, the solitary wave will show fewer structural changes propagating through the inhomogeneous plasma medium. But in the case of Figure 2, we have seen that as the number density of dust particles increases, the width DA solitary wave increases. This result indicates that the solitary wave will propagate with a larger area than the soliton amplitude. The solitary wave will propagate slowly throughout the plasma medium during this propagation. When the electron and ion distribution is relatively high with the higher dust number density, the phase of the solitary wave gets boosted. In this case, the solitary wave will propagate at a higher velocity than in the other cases. From Figure 3, it can also be said that the dust acoustic solitary wave will propagate faster with higher amplitude and higher velocity with the increasing value of $\gamma$ and $n_{d0}$.

Figure 2.

Variations in DAS width w.r.t. number density of dust grains ($n_{d0}$).

Figure 3.

Variations in phase velocity w.r.t. $\gamma$ and number density of dust grains ($n_{d0}$).

9. Conclusion

In this work, we have presented a brief study on the fundamental properties of nonlinear DA wave structure, the interaction of dust and plasma particles, and the effect of intramolecular attraction and repulsion between these plasma and dust grain particles in inhomogeneous dusty plasma. During this investigation, we have discussed a few fundamental relations of dusty plasma, the effect of dust particles on the nonlinear wave structures, and the basic characteristics of DA waves, which exist in both strongly and weakly coupled dusty plasmas. Here, we have also studied the various aspects of dust-acoustic solitary waves (DASWs) in an inhomogeneous plasma. For studying and analyzing the various aspects of DAWs in inhomogeneous plasmas, the governing fluid equations of plasmas are considered to derive the Korteweg-de Vries (KdV) equation. The solution of the KdV equation is obtained as a soliton or solitary wave. The solitary wave solution indicates the various characteristics of DASWs in the inhomogeneous dusty plasma. The above results suggest that the number density of dust particles in DA solitary waves is affected by the negative potentials only. Indeed, we tried to exhibit the effects of dust particles on the nonlinear DA wave structures in inhomogeneous plasma environments. We have also tried to provide a brief idea of dusty plasmas, which is not only fundamental particle of the universe but also dusty plasmas are fundamental concepts like protons and electrons. This chapter also includes a systematic and extensive study on DAWs for inhomogeneous and unmagnetized plasmas.