Propagation of Nonlinear Dust-Acoustic Waves in a Self-Gravitating Collision Magnetized Dusty Plasma in Earth’s Magnetosphere

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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 defnitive 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. Tis 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 lowerenergy states, or it can be the ground state, by emitting radiation over a wide range of the spectrum. Tus, we fnd that the presence of the magnetic feld in the various astrophysical plasma environments has a very efective 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 feld in plasma environments, it was observed that when gravitational waves propagate in the various cosmic environments, they interact with a small magnetic feld, and thus, they produce electromagnetic radiation [3,4]. With many different types of plasma medium, we found that the magnetic feld exerts a force known as the Lorentz force that afects the moving charged particles, which penetrates or diverts the transport path of the charged particles into a spiral, and this, in turn, afects the properties and the formation of nonlinear structures in this manner [5][6][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. Terefore, 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][9][10]. Tus, it happens that very large dust grains are exposed to other types of forces that afect them, for example, both electrostatic forces as well as gravitational forces while the efect on both ions and electrons is sometimes tested by electrical force only because their masses are much smaller compared to the mass of dust. Tus, we fnd that the instability expressed by jeans has a very important and efective role in the phenomenon of the collapse of dusty grains as well as dusty molecular clouds [11].
Since most astrophysical phenomena contain strong magnetic felds and inhomogeneous equilibrium densities, it is important to consider the study of electrostatic waves of self-gravitational ferromagnetic gases that can be classifed as homogeneous or heterogeneous. In the stable Pinnjjmal clouds, we fnd a dusty plasma probe study that can be described in self-gravitational magnetic felds [12][13][14].
Te propagation phenomenon and properties of ionacoustic waves were studied in a nonmagnetic electron-ion plasma, which is characterized by a non-Maxwellian confned 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 feld equation. Where they found that the efect 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 efects 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. Te 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 efect and thus opposes the a conservation law in the system. Tus, in this case, the shock may be a generated wave that occurs due to the strong efect of anomalous dissipation. Knowing the proof, Hirota's simplifed 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]. Te 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 modifed Korteweg-de Vries-Burgers equation. Te nonlinear analysis of the single ion acoustic solutions as well as the nonwide ground plasma shock wave solutions was investigated under the modifed Korteweg-de Vries-Burgers equation where the diferent patterned solutions of the MKdV equation were derived directly from the corresponding Hamiltonian of the system, using the weighted residual method. Tus, approximate analytical solutions of the MKdVB equation are explored using the solution of the MKdV equation as an initial solution [17]. Te 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 efect of the physical factors present in the plasma system on wave propagation in the framework of Schamel Burgers medium [18]. Te 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. Te efect of collisions between particles is neglected during studies of wave dynamics in a dusty plasma environment although these efects may have a signifcant impact on wave formation. For the frst time, a large collision efect 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 efect of the damping coefcient has been described from the point of view of numerical analysis. Tis is in addition to examining the efects 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 diferential 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 diferential equations (PDEs), such as the nonlinear Schröinger equation (NLSE), Korteweg-de Vries equation (KdV), Kadomtsev-Petviashvili equation, and Sine-Gordon equation [20][21][22][23][24]. Tese methods, in fact, are able to express the solutions of integrable diferential equations and obtain various waves when one of these analytical methods is used to obtain the solution. Te physical phenomena in which the above fully integrable nonlinear diferential equations (PDEs) arise tend to be very idealistic in dealing with those equations. Terefore, including infuences 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 efects would imply that the PDE is no longer fully integrable, and hence, it is important to defne the conditions under which the perturbed PDE is fully integrable in order to be able to describe and study diferent physical phenomena [25,26].
Te complex Ginzburg-Landau (CGLE) equation is a nonlinear diferential equation rich in a number of solutions that contain critical values, so it has an efective and essential role in understanding nonlinear wave physics in many nonequilibrium phenomena, especially in dusty plasma physics. A review of specifc 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][30][31].
Motivated by these theoretical works are the following: (1) Investigate the stability of dust sound waves in the Earth's magnetic feld 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 feld Te 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 fuid, 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 efects of the plasma parameters in the model, and the role they play in infuencing 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.

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. Te magnetic feld efect is considered in the z-direction only. Te nonlinear dynamic process of such kind of disturbances is governed by the dust grain beam fuid equations [32,33] where Ψ and η 0 denote the dust shear viscosity and the bulk viscosity coefcient, respectively. We thus consider the densities of both electron and ion numbers to be the Maxwellian distribution, respectively, Te total pressure contains the pressure of the electron-ion gas and the pressure of thermal radiation together [32] Te equation for magnetic induction is given by where the current density is Poisson's equation of gravitational potential is given by Te DA wave potential φ is obtained from Poisson's equation At the equilibrium state, we have the quasineutrality condition that can be expressed as n i0 � n e0 + Z d0 n d0 , where Z d0 denotes the average number of electrons present on a grain of dust, while n d0 denotes the number density of undisturbed dust and fnally φ g denotes the gravitational potential. Where the three-dimensional Cartesian coordinate system is ∇ � (z/zx, z/zy, z/zz), n i0 and n e0 denote the undisturbed number densities of both ions and electrons, respectively, B is the magnetic feld in the direction of wave propagation z, i.e., (B � B z z + 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 (n d0 Z d0 ), as follows: where n d � n d /(n d0 Z d0 ) refers to the normalized density of the number of dust grains in the perturbed state, n e � n e /(n d0 Z d0 ) refers to the normalized density of the number density is the normalized number density of hot electrons in the perturbed state, n i � n i /(n d0 Z d0 ) is the normalized perturbed number density of ions, and u d is the velocity of the negative dusty grains, which is normalized by the dust-acoustic speed C d � (λ D ω pd ). Te electrostatic potential force ϕ and φ g the gravitational potential force are normalized by (K B T eff )/e. Te space coordinates are normalized by the Debye length (λ D � (K B T eff / (4πe 2 Z d0 n d0 )) 1/2 ), and the time (t) is normalized by the inverse of dusty plasma frequency ω − 1 Advances in Astronomy Z 2 d0 e 2 ) 1/2 while the quantities without units of measure that appear in the dusty plasma model after normalization are w ω pd ), β 1 is the damping coefcient 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 coefcient of damping in the dusty plasma system, where bulk viscosity is a material property that determines how a plasma responds to compression [15][16][17][18][19] where d w < λ D is the dust grains radius.
In order to investigate studying propagation in threedimensional envelope dust-acoustic waves, we employ the standard reductive perturbation technique [23] to reduce the basic set of the fuid equations (10)- (19) to one an evolution equation. Consider strongly magnetized plasma and the wave propagates in the z direction with weak transverse perturbations.
We can stretch the independent variables in the dusty plasma model using the following stretched [23]: where ε is the power of arranging the perturbed and greater than zero, and V g is the group velocity of the proliferating dust-acoustic wave. Te dependent variables in the dusty plasma model are expanded as follows: where ω is the angular frequency, and k 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 frst order in ε gives, namely, the frst harmonic mode of the carrier wave (i.e., m � 1 and L � 1). we get the following relations: where b 1 , b 2 , b 3 , and b 4 are given in the appendix. From the perturbed frst order of Poisson's equation, we obtain a linear equation, which is the linear dispersion relation in the dust plasma model under study. where Te linear dispersion relation in equation (28) has two roots for real values of k. Te second-order (m � 2), reduced the equations with harmonic modes (L � 1), we get Advances in Astronomy where b 5 , b 6 , b 7 , and b 8 are given in the appendix. An explicit compatibility (consistency) condition is fulflled via the following relation: Te compatibility condition in equation (34) is exemplifed by the group velocity of the dust-acoustic waves without the vector sign, which is defned in terms of frequency. It is seen that the group velocity is composed of real (V gr ) and imaginary (V gi ) parts, where V gr and V gi be in the appendix where the second harmonic modes (m � L � 2) arising from the nonlinear self-interaction of the carrier dust-acoustic waves are obtained in terms of (φ (1) 1 ) 2 as where b 9 , . . ., and b 15 are given in the appendix. Te 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 L � 0 components of the second order and the reduced equations of the third order, which can be expressed as a function of |φ (1) 1 | 2 as where b 16 , . . ., and b 22 are given in the appendix. Finally, the third-harmonic modes (m � 3 and L � 1), with the aid of equations (27)- (38), give a set of equations. Te compatibility condition for these equations yields the type of the 3D-CGL equation where P 1 is the dispersion coefcient, P 3 is the nonlinear coefcient, P 4 is the dissipative coefcient, and the coefcient P 2 is given in the appendix and φ ≡ φ (1) 1 , for simplicity.

Analytical Solutions of the Dissipative Dust-Acoustic Waves
In this part, we will apply the modifed 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 P 1 � R 0 + iR 11 , P 2 � R 2 + iR 22 , and P 3 � (R 3 + iR 33 ).
Assume the solution in a travelling waveform as follows: where Y � L 1 ξ + L 2 η + L 3 ζ, where L 1 , L 2 , and L 3 are the direction cosine in the coordinates ξ, η and ζ and satisfy the relation L 2 1 � 1 − L 2 2 + L 2 3 , and Ω is the constant determined later. By inserting equation (41) into equation (40) and separating real and imaginary terms, we obtain where where g 1 � d 6 − d 3 , g 2 � d 1 + d 4 , and g 3 � d 2 − d 5 .
Apply the homogeneous balance principle between the nonlinear term and a dispersion term of equation (44) and assume the solution as [36] where c 0 , c 1 , and c 2 are the constants determined later, and G(Y) satisfes the following Riccati equation: where c 3 and c 4 are the arbitrary constants. By inserting equation (45) into equation (44) and collecting a power of G(Y), we get a system of algebraic equations, and solving this system, we get the given constants given the physical parameters such as Te Riccati equation has solutions in references [20,34,37,38].
(1) Te hyperbolic functions solution where c 3 , c 4 , . . ., and c 10 are the arbitrary constants, satisfying the above conditions

Numerical Results and Discussion
In this section, frst, 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 afected 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 L 3 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 fxed 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 n e0 decreases, it in turn leads to the emergence of the soliton wave.

Soliton Structures.
In this subsection, we are interested to investigate the impact of diferent 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 Y for the variation of diferent physical parameters, and it is discussed below in brief: the constraint of the soliton structures, when L 3 is large where this is the direction of propagation of the wave, and this leads to L 1 and L 2 being smaller. Figure 1(a) shows the efect of the energy of the electrons on the amplitude and energy width of the dust soliton acoustic wave. We fnd 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 signifcantly afects the amplitude and energy width of the dust-acoustic soliton wave. We fnd 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 σ e of the relative electrons increases, it signifcantly afects the amplitude and energy width of the dust-acoustic soliton wave. We fnd 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 k decreases, it greatly afects the amplitude and power width of the acoustic dust wave. We fnd that this potential energy increases signifcantly and also leads to an expansion in the width of the dusty soliton wave. In other words, when the wave number k 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 (EMR) values are lower, and the frequency is also lower.

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)). Te constraint of the kink wave structures, when L 3 is smaller where this is the direction of propagation of the wave in a very narrow trajectory, leads to L 1 and L 2 being larger or when the unperturbed number density of electrons is greater than from the frst case, i.e., n e0 � 0.12 cm − 3 . Tis trajectory is clear that the higher temperature ratio σ e 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). Tis 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 σ er 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 k leads to an increase in the energy of the dust wave, and its width is also clearly larger.

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)). Tese waves can be considered the electromagnetic waves, and these vector felds 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 σ e 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 k increases, that is, when the wavelength decreases, the energy of the electromagnetic wave decreases because the frequency decreases.
Te dust-acoustic periodic travelling wave, when the value cosine in the direction of the wave propagation is large, appears, i.e., L 3 � 0.72. Figure 4(a) shows the periodic sharp wave type, and the efect 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 efect of the electron radiation temperature ratio on the behavior of the 8 Advances in Astronomy periodic sharp wave, we fnd that when the electron radiation temperature ratio increases, it leads to a decrease in the wave amplitude.

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 feld. 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. Tis means that when the wave propagation direction is large, which is also the direction of the magnetic feld, 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 feld, the shock-like wave appears. Te 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.