Abstract
The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.
1. Introduction
Wave propagation and instability in magnetic dusty plasmas have been studied extensively in the past few decades since the presence of charged dust particles is so massive that it plays a definitive role in understanding and interpreting electrostatic disturbances in space plasma environments as well as in laboratory plasmas. One of the main properties of gases in plasma conditions is the transfer of radiant energy. This is because it is considered to be an obvious consequence of the excitation of high-energy states of elementary particles in the plasma and the way they revert to lower or lower-energy states, or it can be the ground state, by emitting radiation over a wide range of the spectrum. Thus, we find that the presence of the magnetic field in the various astrophysical plasma environments has a very effective role in modifying the properties and behavior of nonlinear wave propagation in these media [1, 2]. Among the exciting studies conducted in the presence of the magnetic field in plasma environments, it was observed that when gravitational waves propagate in the various cosmic environments, they interact with a small magnetic field, and thus, they produce electromagnetic radiation [3, 4]. With many different types of plasma medium, we found that the magnetic field exerts a force known as the Lorentz force that affects the moving charged particles, which penetrates or diverts the transport path of the charged particles into a spiral, and this, in turn, affects the properties and the formation of nonlinear structures in this manner [5–7]. Recently, it has also been established for the theoretical study, and for dusty plasma physics, there is an unstable spectrum of genes in nonmagnetic and self-gravitational dust gases. Therefore, it has been shown that in dusty nonmagnetic gases, instability of densities occurs from the presence of ion dust-acoustic waves or dust-acoustic waves [8–10]. Thus, it happens that very large dust grains are exposed to other types of forces that affect them, for example, both electrostatic forces as well as gravitational forces while the effect on both ions and electrons is sometimes tested by electrical force only because their masses are much smaller compared to the mass of dust. Thus, we find that the instability expressed by jeans has a very important and effective role in the phenomenon of the collapse of dusty grains as well as dusty molecular clouds [11].
Since most astrophysical phenomena contain strong magnetic fields and inhomogeneous equilibrium densities, it is important to consider the study of electrostatic waves of self-gravitational ferromagnetic gases that can be classified as homogeneous or heterogeneous. In the stable Pinnjjmal clouds, we find a dusty plasma probe study that can be described in self-gravitational magnetic fields [12–14].
The propagation phenomenon and properties of ion-acoustic waves were studied in a nonmagnetic electron-ion plasma, which is characterized by a non-Maxwellian confined electron distribution, in addition to a kappa distribution function with a Schamel distribution. Where the reductionist perturbation theory was applied to obtain the nonlinear and nonplanar Schamel Burgers equation using the basic field equation. Where they found that the effect of viscosity, may cause collisions that lead to the emergence of anomalous dissipation, which usually results in a wave solution of the shock waves. It was found from the investigation in this study that the effects of both viscosity and collisions may cause thumping dissipation [15]. In a dusty, nonmagnetic collisional plasma containing negatively charged dust grains, positive ions, neutral particles, and Maxwellian electrons. The solution was obtained, and shocks were found for the propagation of dust ionic sound waves. Furthermore, assuming that a conservation law is applied in the system, they obtain an approximate solution for the solitary wave, although the existence of the Burgers term causes an increase in the viscosity effect and thus opposes the a conservation law in the system. Thus, in this case, the shock may be a generated wave that occurs due to the strong effect of anomalous dissipation. Knowing the proof, Hirota’s simplified bilinear method was used, and the waveform solution known as shock type was obtained. Again, when the dissipation was weak, the balance between dispersion and nonlinearity could result in a singular solution. Taking into consideration, the case of weak dissipation, the single-wave solution was explored directly without taking into account the conservation law by applying a residual weighted method [16]. The nonlinear analysis of the solitary ion acoustic solutions as well as the shock wave solutions of the nonwide earth plasma has been investigated under the modified Korteweg-de Vries-Burgers equation. The nonlinear analysis of the single ion acoustic solutions as well as the nonwide ground plasma shock wave solutions was investigated under the modified Korteweg-de Vries-Burgers equation where the different patterned solutions of the MKdV equation were derived directly from the corresponding Hamiltonian of the system, using the weighted residual method. Thus, approximate analytical solutions of the MKdVB equation are explored using the solution of the MKdV equation as an initial solution [17]. The propagation of EA waves in viscous plasmas is described and monitored taking into account the weak damping (by adding a Burger term) due to interparticle collisions and viscosity. Particular attention was paid to studying the effect of the physical factors present in the plasma system on wave propagation in the framework of Schamel Burgers medium [18]. The propagation properties of dust, ions, and acoustic waves in nonmagnetic dusty plasma, which consist of mobile ions, negatively charged dust particles, and also trapped non-Maxwellian electrons, as well as both the kappa distribution function and the Schamel distribution, were investigated and studied together. The effect of collisions between particles is neglected during studies of wave dynamics in a dusty plasma environment although these effects may have a significant impact on wave formation. For the first time, a large collision effect was developed in the nonplanar Schamel framework, and by using the conservation law, the approximate analytical solution of the NDS equation was derived. Also, the important effect of the damping coefficient has been described from the point of view of numerical analysis. This is in addition to examining the effects of other physical parameters on the propagation of dust waves in NDS media [19].
Recently, there has been a great interest in studying ways to obtain solutions to nonlinear differential equations (PDEs) that describe the various physical plasma phenomena. For examples of these methods are, the inverse scattering method and a generalized exponential method of solution of several physically interesting, nonlinear partial differential equations (PDEs), such as the nonlinear Schröinger equation (NLSE), Korteweg-de Vries equation (KdV), Kadomtsev-Petviashvili equation, and Sine-Gordon equation [20–24]. These methods, in fact, are able to express the solutions of integrable differential equations and obtain various waves when one of these analytical methods is used to obtain the solution. The physical phenomena in which the above fully integrable nonlinear differential equations (PDEs) arise tend to be very idealistic in dealing with those equations. Therefore, including influences such as damping, external forces, and an inhomogeneous medium in a dusty plasma (e.g., a medium with variable density or depth) may provide a more realistic model to explain those phenomena whereas the inclusion of these perturbation effects would imply that the PDE is no longer fully integrable, and hence, it is important to define the conditions under which the perturbed PDE is fully integrable in order to be able to describe and study different physical phenomena [25, 26].
The complex Ginzburg–Landau (CGLE) equation is a nonlinear differential equation rich in a number of solutions that contain critical values, so it has an effective and essential role in understanding nonlinear wave physics in many nonequilibrium phenomena, especially in dusty plasma physics. A review of specific physical systems which are described by this equation can be found in [27, 28]. However, the physical states are described by the two- and three-dimensional CGLE emerge frequently. It is therefore of attention to study the characteristics solutions of the multidimensional CGLE. In other words, CGLE is the amplitude equation suitable for describing the slow dynamics in the supercritical transition to unidirectional travelling waves [29–31].
Motivated by these theoretical works are the following:(1)Investigate the stability of dust sound waves in the Earth’s magnetic field condition(2)Determine which condition gives rise to both the Soliton wave and the periodic wave when the same solution is used to obtain both waves(3)Determine which condition gives rise to both the shock-like wave and the periodical wave, and also when the same solution is used to obtain them(4)Describe the magnetosonic waves that may appear in Earth’s magnetic field
The following is a summary of this work. In Section 2, we give the relevant dynamical equations, i.e., the equations for the negatively charged dusty plasma fluid, put it into a dimensionless form, and give the electrons and ions associated with them by the equation of neutrality. In Section 3, the standard reductive perturbation theory is used to derive the nonlinear evolution equation, which describes the system of dusty plasma. In Section 4, where two analytical solutions are presented to the evolution equation, and from these solutions, we obtain solutions for the nonlinear dust sound waves of CLGE. In Section 5, we investigate the effects of the plasma parameters in the model, and the role they play in influencing the behavior of the dust-acoustic waves. Finally, Section 6, we obtained a summary of the conclusions about the nonlinear waves describing the dusty plasma system.
2. Basic Equations and Formulation
We consider three-dimensional nonlinear, self-gravitational, magnetization, and collision nonlinear electrostatic wave propagation consisting of three components, namely, dust grain beam, electrons, and ions following the Boltzmann distribution. The magnetic field effect is considered in the -direction only. The nonlinear dynamic process of such kind of disturbances is governed by the dust grain beam fluid equations [32, 33]where and denote the dust shear viscosity and the bulk viscosity coefficient, respectively. We thus consider the densities of both electron and ion numbers to be the Maxwellian distribution, respectively,
The total pressure contains the pressure of the electron-ion gas and the pressure of thermal radiation together [32]
The equation for magnetic induction is given bywhere the current density is
Poisson’s equation of gravitational potential is given by
The DA wave potential is obtained from Poisson’s equation
At the equilibrium state, we have the quasineutrality condition that can be expressed as , where denotes the average number of electrons present on a grain of dust, while denotes the number density of undisturbed dust and finally denotes the gravitational potential. Where the three-dimensional Cartesian coordinate system is , and denote the undisturbed number densities of both ions and electrons, respectively, is the magnetic field in the direction of wave propagation , i.e., , denotes the Boltzmann constant, denotes the magnitude of the electron charge, and and are the electron and ion temperatures where is radiation constant [32, 34].
3. Derivation of the CGL Equation
Let us now consider the system of negative dusty plasma in three-dimensional nonlinear DAWs propagating along the axis in a Cartesian coordinate system. The dynamics of the DAWs can be described by the nonlinear fluid equations [32]and the generalized viscoelastic dust momentum equation
Equations (10) and (18) are closed by the 3-D Poisson equation as
We can normalize the physical quantities in the dusty plasma model, which are the number densities relative to the number density of dust grains in the initial state , as follows: where refers to the normalized density of the number of dust grains in the perturbed state, refers to the normalized density of the number density is the normalized number density of hot electrons in the perturbed state, is the normalized perturbed number density of ions, and is the velocity of the negative dusty grains, which is normalized by the dust-acoustic speed . The electrostatic potential force and the gravitational potential force are normalized by . The space coordinates are normalized by the Debye length , and the time is normalized by the inverse of dusty plasma frequency while the quantities without units of measure that appear in the dusty plasma model after normalization are , , , , , . , is the damping coefficient in the dusty plasma system, where the shear viscosity is a physical property that depends on how the plasma responds to shear stress or the damping process occurs in the plasma. is also the another coefficient of damping in the dusty plasma system, where bulk viscosity is a material property that determines how a plasma responds to compression [15–19] where is the dust grains radius.
In order to investigate studying propagation in three-dimensional envelope dust-acoustic waves, we employ the standard reductive perturbation technique [23] to reduce the basic set of the fluid equations (10)–(19) to one an evolution equation. Consider strongly magnetized plasma and the wave propagates in the direction with weak transverse perturbations.
We can stretch the independent variables in the dusty plasma model using the following stretched [23]:where is the power of arranging the perturbed and greater than zero, and is the group velocity of the proliferating dust-acoustic wave. The dependent variables in the dusty plasma model are expanded as follows:where is the angular frequency, and is the real variable wave number. We take the remainder that all functions in the model satisfy the reality condition. By substituting equations (21)–(26) into the basic set of equations (10)–(20), the first order in gives, namely, the first harmonic mode of the carrier wave (i.e., and ). we get the following relations:where , and are given in the appendix. From the perturbed first order of Poisson’s equation, we obtain a linear equation, which is the linear dispersion relation in the dust plasma model under study.where
The linear dispersion relation in equation (28) has two roots for real values of . The second-order , reduced the equations with harmonic modes , we getwhere , and are given in the appendix. An explicit compatibility (consistency) condition is fulfilled via the following relation:
The compatibility condition in equation (34) is exemplified by the group velocity of the dust-acoustic waves without the vector sign, which is defined in terms of frequency. It is seen that the group velocity is composed of real () and imaginary () parts, where and be in the appendix where the second harmonic modes arising from the nonlinear self-interaction of the carrier dust-acoustic waves are obtained in terms of aswhere , , and are given in the appendix. The nonlinear self-interaction of the carrier dust-acoustic wave also leads to the creation of a zeroth-order harmonic where its strength is determined analytically by taking the components of the second order and the reduced equations of the third order, which can be expressed as a function of aswhere , , and are given in the appendix. Finally, the third-harmonic modes ( and ), with the aid of equations (27)–(38), give a set of equations. The compatibility condition for these equations yields the type of the 3D-CGL equationwhere is the dispersion coefficient, is the nonlinear coefficient, is the dissipative coefficient, and the coefficient is given in the appendix and , for simplicity.
4. Analytical Solutions of the Dissipative Dust-Acoustic Waves
In this part, we will apply the modified extended simple equation technique to get dust-acoustic wave solutions. Since equation (38) is complex, this equation can be converted into exactly the same form as follows [33, 35, 36]:where , , and . Assume the solution in a travelling waveform as follows:where , where , , and are the direction cosine in the coordinates and and satisfy the relation , and is the constant determined later. By inserting equation (41) into equation (40) and separating real and imaginary terms, we obtainwhere , , , , , and . By combining the equal degree in equation (42) and equation (43) in one equation,where , , and .
Apply the homogeneous balance principle between the nonlinear term and a dispersion term of equation (44) and assume the solution as [36]where , , and are the constants determined later, and satisfies the following Riccati equation:where and are the arbitrary constants. By inserting equation (45) into equation (44) and collecting a power of , we get a system of algebraic equations, and solving this system, we get the given constants given the physical parameters such as
The Riccati equation has solutions in references [20, 34, 37, 38].(1)The hyperbolic functions solutionwith , , and .(2)The trigonometric functions solutionwith , and ,(3)The exponential functionwhere , , and are the arbitrary constants, satisfying the above conditions
5. Numerical Results and Discussion
In this section, first, we conduct an analytical study on the behavior of the wave solution in equation (51) using one of the constants (47)–(50) and how it is affected by the physical quantities present in the dusty plasma system.
In our present investigation, we consider the role that physical quantities present in a dusty plasma system play in the mode in which the type of solution wave appears. As the direction of wave propagation changes, that is, when the direction of wave propagation decreases, this leads to the emergence of another type of wave behavior, which is the shock-like wave whereas, when the direction of wave propagation is large, the solitary wave appears, that is, when the physical values in the system are fixed and the value of the direction of wave propagation decreases. It also happens that when the unperturbed number density of electrons increases, it in turn leads to an increase in the unperturbed number density of ions through the state of neutralization, and thus, the shock-like wave appears. While the opposite is that when the unperturbed number density of electrons decreases, it in turn leads to the emergence of the soliton wave.
5.1. Soliton Structures
In this subsection, we are interested to investigate the impact of different compositional parameters on the dust-acoustic solitary waves (DASW) propagated in the considered dusty plasma medium by using the solution (51). To do so, we have analysed the negative solitary potential versus the space coordinate for the variation of different physical parameters, and it is discussed below in brief: the constraint of the soliton structures, when is large where this is the direction of propagation of the wave, and this leads to and being smaller.
Figure 1(a) shows the effect of the energy of the electrons on the amplitude and energy width of the dust soliton acoustic wave. We find that when the energy of the electron increases, this leads to an increase in the potential energy of the dusty soliton acoustic waves and also a slight expansion in the width of the soliton wave. Figure 1(b) shows that when the ion energy increases, it significantly affects the amplitude and energy width of the dust-acoustic soliton wave. We find that when the energy of the ions increases, this leads to an increase in the potential energy of the dust-acoustic soliton waves and also an expansion in the width of the soliton wave greater than in the case of increasing the energy of the electrons. Figure 1(c) shows that when the electron energy, i.e., the temperature ratio of the relative electrons increases, it significantly affects the amplitude and energy width of the dust-acoustic soliton wave. We find that when this energy increases, this leads to a decrease in the potential energy of the dusty soliton sound waves and also contract in the width of the soliton wave. Figure 1(d) shows that when the wave number decreases, it greatly affects the amplitude and power width of the acoustic dust wave. We find that this potential energy increases significantly and also leads to an expansion in the width of the dusty soliton wave. In other words, when the wave number increases, this leads to a decrease in the wavelength and thus increases the radiated energy, i.e., as the wavelength involved increases, the radiated energy tends to be, the values are lower, and the frequency is also lower.
(a)
(b)
(c)
(d)
5.2. Shock-like Structures (Kink Wave)
When inserting the solution (52) in the travelling wave (41), obtaining the kink wave and taking into account the values of the constants ((47), (47), or (49)). The constraint of the kink wave structures, when is smaller where this is the direction of propagation of the wave in a very narrow trajectory, leads to and being larger or when the unperturbed number density of electrons is greater than from the first case, i.e., . This trajectory is clear that the higher temperature ratio of the electrons increases the wave energy of kink wave and its width as shown in Figure 2(a). Figure 2(b) shows the path of the energy of the zigzag wave and its width. It is clear that the higher temperature ratio of the ions increases the energy of the dust wave, and its width is also greater than it is in the case of increasing the energy of the electrons as shown in Figure 2(a). This indicates that when the thermal energy of the ions increases, this leads to an increase in the energy wave. Figure 2(c) shows the path of the dust wave energy and width, and it is clear that the higher temperature ratio of the radiating electrons reduces the energy of the dust wave, and its width is also clearly larger. Figure 2(d) shows the path of the kink wave energy and width, and it is clear that increasing the wavenumber leads to an increase in the energy of the dust wave, and its width is also clearly larger.
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(b)
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5.3. Periodic Wave Structures
Introducing the solution (53) into the travelling wave (41), we obtain another type of waveform solution, which gives us a type of periodic wave, given the values of the constants ((47), (47), or (49)). These waves can be considered the electromagnetic waves, and these vector fields have a sine waveform, are oriented at right angles to each other, and oscillate perpendicular to the direction of dust-acoustic wave travel (Figure 3). It is clear that when the temperature ratio increases in the dusty plasma system, the nonlinearity in the system decreases, and this leads to a decrease in the energy of the electromagnetic wave and a decrease in the width of that wave as shown in Figure 3(a). Figure 3(b) shows that when the wave number increases, that is, when the wavelength decreases, the energy of the electromagnetic wave decreases because the frequency decreases.
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(b)
The dust-acoustic periodic travelling wave, when the value cosine in the direction of the wave propagation is large, appears, i.e., . Figure 4(a) shows the periodic sharp wave type, and the effect of the electron temperature ratio on the behavior of the periodic wave. where the electromagnetic potential energy increases, when the electron temperature ratio increases. While Figure 4(b) shows the effect of the electron radiation temperature ratio on the behavior of the periodic sharp wave, we find that when the electron radiation temperature ratio increases, it leads to a decrease in the wave amplitude.
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(b)
6. Summarize
We investigated the properties of physical parameters on a dust-acoustic wave propagation of electrostatic dust in magnetized plasma containing isothermal electrons and hot ions in the presence of a magnetic field. Standard reductionist perturbation theory is used to derive the corresponding 3D-CGL equation that governs the dynamics. One of the useful results is the link between the emergence of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic carrier waves. This means that when the wave propagation direction is large, which is also the direction of the magnetic field, the soliton wave appears whereas when the wave propagation direction is small, that is, the farther it is from the perpendicular direction of the magnetic field, the shock-like wave appears.
The advantage of using the method under study over the standard method is that when we used the harsh method, we get two waveforms using the same solution [39, 40], but here we get three waveforms for the same solution.
The present work applies to understanding the propagation and formations of dust waves in the attractive particle dust cloud when radiation pressure and strong coupling effects are simultaneously present in the dusty plasma system. The results can also be useful in understanding the nonlinear propagation of acoustic waves of dispersed dust in the viscomagnetic viscosity laboratory that has been used in the fundamental study of nonlinear dust activities in the dust cloud and astrophysical consequences.
Appendix
A: Constants
B: The solution of the Riccati equation is
, we suppose thatis the solution of equation (B.1) where
The characteristic equation iswherewhere
The general solution is
When , this tends to , and the general solution is
When , this tends to , and we define , , , and
The two solutions of the differential equation are
The general solution is
When , this tends to , and we define , , and the general solution iswherewhere .
Data Availability
Data sharing applies to this article in references [30–32] as the datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to complete this work.
Acknowledgments
The authors express their gratitude to Princess Nourah Bint Abdulrahman University Researchers Supporting Project number PNURSP2023R2, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.