STRUCTURAL VARIATIONS OF DUST ACOUSTIC SOLITARY WAVES (DASWs) PROPAGATING IN AN INHOMOGENEOUS PLASMA

This paper presents our theoretical investigations on the structural variations of dust acoustic solitary waves (DASWs) in inhomogeneous unmagnetized plasmas. To study the structural variations of DASWs, we have considered collisionless, hot isothermal, and Boltzmannean distribution for electrons-ions with negatively charged dust grains in weakly inhomogeneous plasmas. We have used the reductive perturbation technique (RPT) in the governing equations of plasmas, derived the modified Korteweg-de-Vries (m-KdV) equation, and obtained the solitary wave solution. We have considered the appropriate stretched coordinates for space and time variables for the inhomogeneous plasma. This paper investigates the effects of dust particles on ion-acoustic solitary waves' propagation in the inhomogeneous plasma model. We have also included the effect of inhomogeneity parameters on the soliton structures.


I. INTRODUCTION
Plasma and charged dusts or dust grains are the two fundamental ubiquities of the universe.Dusty plasma is a relatively new branch of the research area in plasma physics.In dusty plasmas, the interaction of dust grains and plasma particles are studied.Plasma particles consist of hot and electrically conductive electrons and ions, but the dusty plasma particles consist of both plasma particles and dust grains, i.e., dusty plasma particles are comprised of three fundamental ingredients ions, electrons, and charged dust particles or dust charged grains.The dust particles show more complex behaviours when added to the plasma particles.So, the dusty plasmas are also known as complex plasmas or multicomponent plasmas.Investigations on these complex or multi-component plasmas are abundant throughout the universe.The majority of extant solid matter is thought to be made up of dust grains that are frequently contained by plasma particles.Planetary magnetospheres, cometary environments, nebulas, etc. [1], are some examples of dusty plasmas.These complex plasmas are enormous important to understand the theories of geophysics better, complete some space missions, advancements of knowledge in astrophysical environments like the formation of stars, galaxies, and nebulae, and manufacture some new materials the in the semiconductor industries, etc. Dusty plasmas are one of the significant fields for working in Controlled Thermonuclear Research (CTR).In 1982, Voyager spacecraft [2] discovered the radial spokes of the Saturn's B ring.Apart from the wide range of applications of dusty plasmas in the astrophysical problems, researchers have observed the wide use of dusty plasmas and their related issues during the use of dusty plasmas in the manufacturing of new materials in the semiconductor industries [3].Later, a number of authors have worked in the both theoretical and experimental works on the dusty plasmas.In the early eighties of last century, the concept of dusty plasma could not be developed convincingly.However, to understand the fundamental properties of dusty plasmas, some devices are introduced in dusty plasma laboratories like the rotating drum system [4] and dust shaker systems [5].Subsequently, in the review literature of Goertz [6] and Northrop [7], we have some details of works carried out in dusty plasma for astrophysical cases.Basically, for unmagnetized plasmas, dusty plasma waves are characterized by three different modes such as dust-acoustic waves [9] (DAW), dust ion-acoustic waves [8] (DIAW), and dust lattice waves [10] (DLW).Several theoretical [8][9][10] and experimental [11,12] studies have been done to achieve more understanding on the complex behaviours of DAWs, DIAWs, and DLWs.Apart from the above, large number of literatures could be found on the various nonlinear wave phenomena and instabilities for homogeneous dusty plasmas, inhomogeneous dusty plasmas [13][14][15][16][17][18][19][20] and nonlinear wave excitation in nonequilibrium plasmas.For studying the linear wave theories in plasmas, the nonlinearities for small amplitude waves are not considered, but in the case of large amplitude waves, the nonlinearities cannot be neglected.Due to the existence of nonlinearities in the plasma waves, the various physical parameters and their effects can be studied.The nonlinearities are also indicated in the experimental and theoretical behaviours of some nonlinear plasma wave structures such as solitons or solitary waves, supersolitons, rouge waves, shock waves, etc.In inhomogeneous plasmas, few more researchers have done their works on the properties and effects of dust ion-acoustic (DIA) solitary waves [21,22], dust acoustic (DA) solitary waves [8,27,28,32], shock waves [23][24][25][26], and dust lattice (DL) solitary waves [10,29], etc. Apart from the broad applications of complex or dusty plasmas in astrophysical systems and space science, the wide applications of dusty plasmas have also been seen in the fusion sciences and laboratory environments [30,31].Gogoi and Deka also studied the propagation of dust acoustic (DA) solitary wave propagations in inhomogeneous plasmas [32].Some more basic properties and the effects on dust-acoustic waves (DAWs) have been studied theoretically in the presence of both positively and negatively dust particles with negatively charged ion fluids and ᴋ-distributed superthermal electrons [33].Dusty plasmas are basically studied in inhomogeneous plasma systems to understand the advanced use of plasmas in astrophysical environments and in laboratory discharges [34].The modulational instability and dust-cyclotron wave for DAWs in three-component plasmas have been explored in the presence of ions, electrons, and dust particles under the influence of a magnetic field [35].In the presence of pair-ion fullerene plasma, magneto-acoustic wave propagation was also studied in both the linear and nonlinear plasma environments [36].Pakzad and Nobahar [39] studied the properties of dust ion-acoustic (DIA) solitary waves in inhomogeneous unmagnetized plasmas in the presence of stationary dust grains, super thermal electrons, and inertial ions.They have also studied the behaviours of DIAWs propagating in the various astrophysical environments like Solar winds, Venus's ionosphere, and the Earth's atmosphere.Dehingia and Deka [41] have recently studied ion-acoustic solitons' variations in an inhomogeneous plasma.They have observed how the ion-acoustic solitary wave bends in some critical points due to the inhomogeneity present in the plasma system.Though many investigations on the effect of dust particles on plasma properties have been done till date, there are still many scopes to study the role of dust particles in affecting the nonlinear structures of plasmas, in particular for the case of inhomogeneous plasmas.In this paper, we will extend the investigations of Gogoi and Deka [32], to understand the structural variations of dust-acoustic solitary waves in inhomogeneous plasma under the following considered physical situations.
Here, we present our investigations on the structural variations of dust acoustic solitary waves in inhomogeneous plasma in the presence of hot isothermal electrons with Boltzmannean electron-ion distribution.This model consists of unmagnetized, collisionless, hot isothermal electrons and weakly inhomogeneous plasmas in the presence of negatively charged dust grains.Using the governing equations of plasmas and the reductive perturbation method or technique (RPT), we have derived the modified Korteweg-de-Vries (mKdV) equation with the help of appropriate stretched coordinates for space and time variables for the inhomogeneous plasmas.The solution of the above mKdV equation also indicates the various nonlinear effects of dust grains propagating in inhomogeneous, unmagnetized plasmas in the presence of negatively charged dust particles.We have also presented our results and investigated the effect of dust particles on ionacoustic solitary waves' propagation due to the inhomogeneity parameters in the inhomogeneous above-considered plasma model.

II. GOVERNING EQUATIONS
We have considered an unmagnetized, collisionless, hot isothermal, Boltzmannean distributed electrons and ions, in the presence negatively charged dust particles in weakly inhomogeneous plasma.With the variable density gradient and along the x-direction only, the system is considered to be inhomogeneous.The set of dimensionless and nonlinear governing fluid equations for slowly moving dust acoustic waves along the x-direction is taken as follows: Continuity equation: Momentum equation: Poisson equation : Boltzmannean distribution of electrons: Boltzmannean distribution of ions: In the above equations,  represents the number density of dust grains which is normalized by  at an equilibrium condition.Here,  represents the fluid velocity of dust grains which is normalized by dust acoustic speed (DAS)  =  where ion temperature is  and mass of the charged dust particles is  .Here, we assume  as the electrostatic potential with the charge neutrality at equilibrium state  =  +   .The quantities  ,  , ,  ,  are reduced to the dimensionless form with the help of Debye length  = , DAS  and thermal electrostatic potential  = where  is the electron temperature and  is the electron charge.For charged dust grains  =  , the balanced current equation is given by [6]  +   =  +  where  and  are dust currents,  and  are the ion and electron currents respectively.Due to slow motion of the dust fluid its velocity  is small at equilibrium state, the dust current is balanced by both the ion and electron currents together.Then we have the balanced current equation is given by [10]  +  ≈ 0 where  and  are respectively given by [40] where  denotes the radius of charged dust grains and  =   .

III. DERIVATION OF MODIFIED K-dV (m-KdV) EQUATION
To study the structural variations of dust acoustic (DA) solitary wave propagations in inhomogeneous plasmas, we use the reductive perturbation technique (RPT).To apply the RPT for some small amplitude wave, we use an appropriate one-dimensional space-time stretched coordinate which is given by [37] where  is a smallness parameter,  is the phase velocity of the DA soliton, normalized by  .Now, using Eq.(7) in Eqs.(1) − ( 5) we get the following set of equations are as follows: Expanding the dependent variables  ,  ,  and  about the equilibrium parts in terms of power series of  as follows: Using the Eq.(11) in Eq. ( 8), ( 9) and (10) respectively we get the following set of equations as follows: (Neglecting the higher terms having powers of  more than ) At  , we get, (  ) = . (16) At , we get, So, using the boundary conditions  ,  ,  ⟶ 0 as || ⟶ ∞, we get from (16) − ( 18) where and where  determines the phase velocity of DAW.At the highest order coefficients of  we get, Now, eliminating all the 2 nd order quantities from Eqs. from (24), ( 25) and ( 27) and adding all of them we get a nonlinear PDE with variable coefficient is of the form: where ,

𝐶 = +
. The above Eq.( 28) is in the new form of the KdV equation known as mKdV equation, as there is an additional term due to the plasma inhomogeneity of number density.The solution of the above modified K-dV (mKdV) equation i.e., Eq. ( 28) represents the structural variations of nonlinear dust acoustic (DA) solitary waves propagating in inhomogeneous plasma.The nonlinear constant coefficient A and the dispersion coefficient B depend on the inhomogeneous number density.The extra term in the above considered inhomogeneous plasma model appears with the coefficient C due to the inhomogeneous number density gradient.

IV. SOLUTION OF MODIFIED K-dV (mKdV) EQUATION
We have considered the transformation 38

𝜙 = 𝜇𝑒
to get the solitary wave solution of the Eq.(28) for the KdV equation is of the form where  =  and  = .The above nonlinear coefficients  and  functionally depends on the environment of the chosen plasma system.To reduce the complexities of the calculations, the variations are considered relatively small compared to the locally constants parameters.Now, we have considered a new frame of reference  =  −  w.r.t.velocity  to solve the Eq.(29).After using this new frame of reference and the Kodama -Taniuti method [38], we have obtained the solution of Eq. ( 29) is as follows: where  = and  = are the width and amplitude of the DA solitary wave respectively.

V. RESULTS AND DISCUSSION
In this paper, we have studied the nonlinear DA solitary waves to understand the structural variations of DA solitary wave while propagating in inhomogeneous, unmagnetized plasmas consisting of negatively charged dust grains.We have used the RPT method to derive an mKdV equation with the variable coefficients.We have considered appropriate stretched coordinates for both space and time to employ the RPT method.Then the Kodama-Taniuti method [38] is applied to get the DA solitary wave solution, and the numerical results for DA solitary wave propagation are obtained in Eq. (30).We have focused on the issues of structural variations of DA solitary waves in the above-considered plasma environment.
It is clear from the Eq.( 23), the phase velocity  depends on the various choices of  ,  ,  , and .We have observed the coefficients  and  in the above Eqs.Based on the above results, it can be observed that with the increasing/decreasing of values of  and n , the shape variations of the soliton amplitude will also be seen.With the increasing/decreasing of values of  and n , the solitary wave amplitude will be increased/decreased, while on the other hand, in Fig. 𝟓, the amplitude of the solitary wave will be decreased/increased with the increasing/decreasing values of n simultaneously.Also, the solitary waves experience more deformation due to the faster compression and vibrations of intermolecular interactions in ions, electrons, and dust particles in the plasma system.So, the DA solitary waves produce variations in the soliton structures while propagating in an inhomogeneous plasma.The variations of structures in the amplitude  and width  of the DA solitary waves also occur with the increasing/decreasing of number density  .While the DA soliton propagates in inhomogeneous plasma, it gets deflected due to plasma of inhomogeneity i.e., the number density  .From the above figures (Fig. 5), it can be observed that with the increase in number density and ion density profile, the smaller the soliton amplitude.Also, the higher the number density, electron density, and ratio of temperature difference, the higher the soliton amplitude which is shown in Fig ( 6, 7).Similarly, when the number density and ion density increase, the width of the solitary wave decreases (Fig. 8) Also, when the number density and electron density increase, the width of the solitary wave is increased Fig ( 9, 10).Thus, it can be ensured that the structure of the solitary waves varies with the variations of amplitude and width of the DA solitary waves.From the above observations, it is observed that the amplitude with the width of the solitary waves slightly deforms during the propagations in inhomogeneous plasma.So, it ensures the shape conservation of solitary structures as the first principle of K-dV soliton.

VI. CONCLUSION
In this work, theoretically, we have studied variations in the structures of DA solitary wave propagations in inhomogeneous unmagnetized dusty plasma.We have considered collisionless, hot isothermal and Boltzmannean distributed ions and electrons, with negatively charged dust grains in weakly inhomogeneous plasmas.In our problem, the basic governing fluid equations are considered and the reductive perturbation technique (RPT) is employed to solve the modified KdV (mKdV) equation.We have used an appropriate set of stretched variable to use the RPT in the governing fluid equations of plasmas.Due to inhomogeneity in the plasma system, an extra term arises in the modified KdV (m-KdV) equation associated with charged dust particles and the inhomogeneous density gradient.During the investigations, we have also studied the structural variations of amplitude and width of the DA solitary waves depending on the various choices of the parameters i.e., number density of the dust grains ( ), electron number density ( ), temperature ratio of ion to electron (), and the ion number density ( ).
In this paper, primarily, we have studied the variations in soliton structures of DA solitary wave propagation in inhomogeneous plasmas.But during the investigations, we have also observed and established a relation between amplitude  and width  of the dust acoustic (DA) solitons shown in the eq.( 30).The Eq. (30) implies that if the plasma inhomogeneity is neglected, the soliton amplitude  will increase with the decrease in soliton width .But based on the various choices of the parameters  and  , the relation between the amplitude  and width , shown in the eq.( 30), has been changed captiously.Also, from the above results, it is clear that  and  increase with the −   +   = 0.
shows the dependency of the phase velocity  with the inhomogeneous number density  and with the various choices of parameters  ,  , , and  .Here, we have introduced the various figures of phase velocity depending on the various choices of parameters  ,  , , and  respectively.From Fig. , we have seen that the phase velocity of the DA solitary wave i.e.,  increases with the increase in the number density  , with the less significance of  .But in Fig. , the phase velocity  increases uniformly with the increasing values of  and  .Similarly, Figs. and  show the expanding rate of  w.r.t. , where the phase velocity  decreases with the increase in .Also Figs. and  also indicates the rapid decrease of the phase velocity with the increasing values of  .

Figure 4 .
Figure 4. Dependency of phase velocity  with  = 1,  = 0.03, and  = 2 (28) and (29), which indicates the dependency of the amplitude  of the DA solitary waves on the parameters  ,  ,  ,  , and .Depending on the various choices of the values of  ,  ,  ,  , and , the structural variations in the amplitude of the DA solitary wave  w.r.t. the density gradient n will be seen in Figs.,  and .Here, the Figs. and  shows the variations in the amplitude of DA solitary wave A w.r.t.n , which is increasing with the increasing values of  and n , respectively.