CHARACTERISTICS OF NONLINEAR DUST ACOUSTIC WAVES (DAWS) PROPAGATING IN AN INHOMOGENEOUS COLLISIONLESS MAGNETIZED DUSTY PLASMA 1

In this paper, we have presented our investigation on the characteristic of nonlinear dust acoustic waves (DAWs) propagating in an inhomogeneous collisionless magnetized dusty plasma (MDP). In this problem, we have considered a collisionless plasma consisting of nonthermal ions, non-extensive electrons and negatively charged dust grains. Using the reductive perturbation theory (RPT) we have derived the modified Zakharov-Kuznetsov (m-ZK) equation. The solution of m-ZK equation indicates the nonlinear characteristics of the DASWs in plasma. Our investigation also predicts how the amplitudes of nonlinear DASWs are significantly modified due to the influence of magnetic field, non-extensive electrons and inhomogeneity parameters in plasma. The results obtained in this investigation may be useful for understanding the propagation characteristics and modification of structures of nonlinear waves in both laboratory and astrophysical plasmas.


I. INTRODUCTION
Solitary waves or solitons are nonlinear wave packets which maintain their shapes during their propagation at a particular speed.Solitons occur due to the mutual cancellation of the nonlinear effects and dispersive effects in the medium.Washimi and Taniuti [1] derived the Korteweg de-Vries (KdV) equation to describe the ion-acoustic solitons (IAS).Nishikawa and Kaw [2] discussed the propagation of ion-acoustic solitons in an inhomogeneous plasma.Theoretically, Kuehl [3] discussed the propagation and reflection of ion-acoustic solitons in an inhomogeneous plasma.He has observed the variations in the soliton amplitudes of incident and reflected solitons.Nejoh [4] investigated the effects of ion temperature on the characteristics of soliton propagation in a relativistic plasma.A few plasma physics researchers studied the properties and characteristics of solitary waves propagating in the presence of various physical situations such as negative ions [5,6] and dust grains [7] in the plasma and solitary wave excitation in nonequilibrium plasmas [30][31][32][33].Kakad et al. [8] provided an experimental study on the validity of fluid theory and chain formation of nonlinear wave propagation in a collisional magnetized plasma.Later, using Kundu-nonlinear Schrödinger equation (Kundu-NLS) Shi et al. [9] discussed the dynamics of nonlinear nonlocal solitary wave solutions propagating in an inhomogeneous plasma.Rani and Yadav [10] studied the characteristics of electron acoustic-solitary waves (EASWs) propagating in a dense magnetized collisional plasma in the presence of degenerate quantum electrons.Recently, Dehingia and Deka [11] have discussed the variations of IAS structures propagating in an inhomogeneous plasma in the presence of hot isothermal electrons.They have observed that at a certain point the structure of IAS gets deformed due to the presence plasma inhomogeneity during their propagation through the system.
Dusty plasma (DP) is a very important research field in plasma physics.DP consists of ions, electrons, and charged dust particles.When the dust particles are included in the plasma, the system indicates some complex behaviours in the system.Thus, DP is also termed as the multicomponent plasma or complex plasma.These DPs are observed in planetary magnetospheres, cometary environments, planetary ring, and nebulas etc. [12].The study of dusty plasma helps us to understand the astrophysical phenomena, the geophysical theories, and importance of space missions etc. Goertz [13] worked on the fundamental properties of dusty plasma in an astrophysical environment.Many researchers studied the basic properties of dust ion-acoustic waves (DIAWs) [14] and dust-acoustic waves (DAWs) [15] propagating in an inhomogeneous plasma.Shukla and Mamun [16] introduced basic structures, properties and propagation of DAWs, DIAWs, dust-cyclotron waves (DCWs) and dust lattice waves (DLWs) etc. in inhomogeneous plasmas.Using the kinetic theory Baluku and Hellberg [17] provided a brief description on the propagation of DIAWs in the presence of  − distributed electrons and negatively charged dust grains in the plasma.Alinejad, and Khorrami [18] studied on the structures of DAWs propagation in the presence of trapped ions and polarized Debye sheath in a strongly coupled inhomogeneous plasma.Atteya, Sultana, and Schlickeiser [19] investigated the effect of superthermal electrons, positive ions as well as negative ions in the propagation of DIAWs in an inhomogeneous magnetized plasma.Akhtar et al. [20] discussed the dynamics of DAWs and DCWs during their propagation in the magnetized plasma.Rehman, Mahmood, and Hussain [21] studied the behaviour of nonlinear magneto-acoustic waves (MAWs) in the presence of warm, collisionless pair-ion (PI) fullerene plasma.In their analysis, they concluded that due to the presence of ion inertial length in the plasma, the effects of wave dispersion are observed in PI plasma.In the study of plasma physics, linear theory is used to study the 505 Characteristics of Nonlinear Dust Acoustic Waves (DAWS) Propagating...

EEJP. 1 (2024)
small amplitude waves without considering the nonlinearities in the plasma.But in the case of large amplitude waves, nonlinearities cannot be ignored.In the plasma studies, nonlinearities play an important role in the nature, properties and characteristics of the dusty wave phenomena.In experimental and theoretical studies, we observe various nonlinear dusty wave structures such as shock waves, rouge waves, solitons, supersolitons etc. Atteya et al. [22] studied the propagation of nonlinear DAWs in an inhomogeneous quantum plasma in the presence of magnetic field.They have observed that the low-frequency longitudinal waves have the potential to trap electrons which obtained from the high-intensity magnetic fields during the modulation of plasma species.Using the Pseudopotential method Pakzad and Nobahar [23] discussed the important properties of DIASWs propagating in an inhomogeneous unmagnetized plasma in the presence of superthermal electrons, inertial ions, and stationary dust grain particles.They have also analysed the modification of nonlinear wave structures of DIAS propagating in an inhomogeneous plasma in the presence of the critical parametric values of superthermal electrons and electron-ion density ratio.Dehingia and Deka [24][25][26][27] studied the various properties of DAWs, modification in DASW structures, effect of dust particles in soliton propagation and propagation of nonlinear waves in the presence of negatively charged dust grains with charge fluctuations in inhomogeneous plasma.There are still many scopes to study on the propagation characteristics of nonlinear wave structures in inhomogeneous plasma depending on the various astrophysical conditions.In this problem, we consider a collisionless magnetized plasma consisting of cold ions, non-extensive electrons and negatively charged dust grains.In this investigation, we have discussed the characteristics of nonlinear DASWs propagating in an inhomogeneous collisionless magnetized plasma in the presence of nonthermal ions with non-extensive electrons and negatively charged dust grains.

II. GOVERNING EQUATIONS
In this article, we have studied the characteristics of nonlinear DASWs propagation in the presence of negatively charged mobile dust grains,  − distributed non-extensive electrons of temperatures  and  , and nonthermal ions with finite temperature  , where  ≪  ≪  in an inhomogeneous magnetized plasma.These temperatures are expressed in the units of energy.The set of governing equations for nonthermal ions, non-extensive electrons and negatively charged dust grains in a dusty plasma system are given by [25][26][27][28][29] ⃗ + ∇ ⃗ .  ⃗ = 0 (1) Now, for equilibrium condition, the charge neutrality equation is given by  =  +  +   where we consider  ,  and  are taken as number densities of low temperature and high temperature electrons and ion number density respectively in the plasma.In the above equations,  is dust charge number,  is the density of dust grains, and  and  are the strengths of non-extensivity of two types electrons with temperatures  and  respectively in the plasma.In this mathematical model,  ⃗ reparents the dust fluid velocity of the dust grain particles normalized by  = ,  is the mass of dust grains, ϕ ⃗ is the electrostatic potential which is normalized by , time  is also normalized by  = and the space variable  is normalized by  = .We have also considered the nonthermal parameters are β = ( ) , where  is the parameter which determines the density population of nonthermal ions in the plasma.The values of the parameters β and  are taken to be 0 ≤ β ≤ and  ≥ 0. In the above equations, we have also considered some set of dimensionless quantities such as  = ,  = ,  = ,  = ,  = , and  = etc.

III. DERIVATION OF MODIFIED ZK-EQUATION
To study the propagation of nonlinear DASWs and their characteristics for small amplitude waves, we use the RPT to derive the m-ZK equation for the plasma system.The solutions of m-ZK equation show the propagation characteristics of the nonlinear DASWs in the plasma.To use the RPT, we use a set of appropriate stretched coordinates [1,[24][25][26][27] for the weakly inhomogeneous plasma is as follows: where the phase velocity  is normalized by  , and the smallness parameter  measures the strength of the dispersion.
Here, all the axes , , and  are normalized by Debye length ( ), and  is normalized by ion plasma period respectively.
To apply the RPT, we use expanded dependent variables  ,  ,  ,  and  along with their perturbed values and in terms of  is as follows [26]: Now, using Eqs.( 7) and (8) into Eqs.(1) − (6), we obtain the 1 st order quantities for  − component of momentum and Poisson's equations are as follows: The above Eq.(10) indicates the phase velocity of the DASWs under the influence of non-extensive electrons, nonthermal ions and negatively charged dust grains propagating in an inhomogeneous MDP.
Similarly, we obtain 1 st order  and  components for the momentum equation are as follows: The above Eqs.(11) and (12) represents the velocities of DASWs in  and  components in the plasma.These two equations also satisfy the 2 nd order continuity equation for DASWs in the plasma.Again, using Eqs.(7) and (8) in Eqs.
(1) − (6) and eliminating the 1 st order terms of  and  components in momentum and Poisson's equation, we get the 2 nd order terms are as follows: The above Eqs.(13) − (15) represents the  and  components of the dust polarization drift.Now, proceeding the same way we will obtain the higher order terms of continuity equation and  − components from momentum equations.Now, eliminating the higher order terms  ,  , and  finally we obtain the equation is as follows: where, The above equation (19) represents the  −  equation which describes the oblique propagation of DASWs in the presence of non-extensive electrons having two distinct temperatures in the MDP.

IV. SOLUTION OF m-ZK EQUATION
To study the characteristics of nonlinear DASWs propagating in an inhomogeneous plasma, we have considered the plasma with an effect of magnetic field at an angle  with Z-axis.Considering the Y-axis fixed, the coordinate axes are assumed to be rotated at an angle  which is significantly modified under the influence of magnetic field, non-extensive electrons and inhomogeneity parameters in the plasmas.Thus, we use a set of transformation equations for the independent variables is as follows: Using the above transformation equations [1], [26], [27], we rewrite the above  −  Eq. ( 19) is as follows: where Now, using the transformation equation [27], the steady state solution for  −  equation is given by where  =  −   and  represents the constant speed normalized by  .Using the above transformation equation, the above  −  equation can be rewritten in steady state form is as follows: Now, using appropriate boundary conditions, i.e.,  → 0, → 0, → 0 as  → ±∞.Then solitary wave solution of Eq. ( 27) is given by where  = and  = represents the amplitude and inverse width of the solitary wave solutions respectively.
Since  > 0, so it is clear from the Eqs.( 19), ( 21) and (24) that based on the sign of , the solitary waves or solitons will only be associated with negative potential ( < 0) of the wave propagation in the plasma.
In the above derivations, we have observed that the Eq. ( 27) is the one-dimensional steady state form of  −  equation.The steady state solution of  −  Eq. ( 27) and standard  −  equation both gives the same results in one-dimensional steady state cases.To obtain the localized solitary wave solution, we use appropriate boundary conditions and transformation equations [27] to solve the Eq.(27).In this problem, we have considered the steady state solution of  −  Eq. ( 19) in one-dimensional cases where all ′s → 0 provided  and  .Thus,  and  will occur in our wave solutions obtained in our above derivations.EEJP. 1 (2024) Hirak Jyoti Dehingia, et al.

V. RESULTS AND DISCUSSIONS
Using the above results and derivations, we have plotted some figures to describe the propagation and characteristics of nonlinear DASWs propagating in an inhomogeneous MDP consisting of nonthermal ions, non-extensive electron with different temperatures and negatively charged massive dust grains.In Fig. 1 and Fig. 2, we have plotted the graphs to show the variations of amplitudes of DASWs w.r.t. and  depending on the various parametric values of non-extensive parameters  and  respectively.Fig. 3 shows the variations of width () of DASWs w.r.t.number density of relative electrons ( ) depending on the various parametric values of nonthermal parameter ().Depending on the various values of the magnetic parameter , the variations of width () of DASWs w.r.t.oblique parameter  are also shown in Fig. 4. Finally, Fig. 5 describes the variations of phase velocities of DASWs w.r.t. the nonthermal parameter () propagating in an inhomogeneous magnetized dusty plasma.In this problem, we have also investigated the effect of magnetic field on the propagation of MDP under the above considered plasma situations.To study the effects of magnetic field and propagation characteristics of DASWs propagating in an inhomogeneous plasma in the nonthermal ions, non-extensive electron with different temperatures and negatively charged massive dust grains, we have used RPT to derive and solve the  −  equation for small amplitude waves by using the various dependent variables of perturbed number densities of ions, electrons and dust grains with dust velocities and electrostatic potentials in the plasma.From the above investigations, Fig. 1 indicate that the magnitudes of amplitudes of DASWs increases gradually with the increase in the temperature ratios ( ) and non-extensive parameter ( ).On the other hand, Fig. 2 shows that the magnitudes of widths of DASWs decreases gradually with the increase in the temperature ratios ( ) and non-extensive parameter ( ).However, in Fig. 3 the number density of relative electrons ( ) and the values of nonthermal parameter () for ions, increases the width of DASWs during their propagation through the plasma.This shows the importance of nonthermal ions which play an important role on stabilizing, forming and the propagation of DASWs in an inhomogeneous plasma with various physical situations e.g., nonthermal ions and non-extensive electrons present in the laboratory and astrophysical plasmas such as protostellar disk, circumstellar and interstellar clouds, cometary tails, Saturn's rings, earth's magnetospheres, solar winds and in asteroid zones etc.In Fig. 4, we have seen that the variations in the oblique parameter  effects the width () of DASWs which increases due to the presence of  with lower angles in between 0 °and 45 °.On the other hand, the width () of DASWs which decreases with higher angles of  in between 45 ° and 90 °.Also, when  → 90 °, the amplitude and width of the soliton tends to ∞ and 0 respectively.This concludes that the waves will not be considered as electrostatic anymore.In this investigation, we have considered the magnetic field parameter as  = , where  and  are dust cyclotron frequency and dust plasma frequency respectively.Fig. 4 also shows that while dust cyclotron frequency i.e.,  increases, the width () of the soliton decreases in the plasma.In the above derivations, we have also derived the dispersion relation and phase velocity in Eq. (10).However, in Fig. 4, it is also observed with the increase in the values of , the amplitudes of the solitary wave structures become relatively spiky and so that the system become destabilized.On the other hand, the width () of DASWs decreases with the increase in the value of external magnetic field parameter .In Fig. 5, we have shown the effects of nonthermal ions and non-extensive electrons and the phase velocity of the DASWs propagating in an inhomogeneous MDP.In Fig. 5, we have also shown the increase of phase velocity of the DASWs slowly and steadily but later it increases rapidly with the increasing values of nonthermal parameter () for ions in the plasma.When the parameters other than  are considered to be constant, the phase velocity of the DASWs increases rapidly with the increase in critical value of .The critical value of nonthermality parameter () depends on the relatively non-extensive electrons present in the plasma.Also, when the electron non-extensivity increases, the nonthermality of ions increases in the modelled plasma which are shown in Fig. 5. From the above discussions, it is clear that the phase velocity of DASWs especially much depend on the nonthermality of ions.However, the phase velocity of DASWs less dependent on the non-extensivity of electrons present in the system.

VI. CONCLUSION
In this paper, we have studied the fundamental properties of DASWs propagating obliquely in an MDP consisting of nonthermal ions, non-extensive with distinct two temperature electrons and negatively charged dust grains.In this problem, we have also discussed the fundamental characteristics of the DASWs propagating in an inhomogeneous plasma under the above considered physical situations.We have observed that with the increase in dust cyclotron frequency, there is an increase in the spikes of amplitudes of DASWs structures in the plasma.The above results also indicate that with the increasing values of  and , the width of DA solitons increases during their propagation through the inhomogeneous collisionless magnetized plasma.We have also observed that the oblique parameter  effects the variations in the width () of DASWs which increases due to the presence of  with lower angles in between 0 °and 45 °.However, the width () of DA soliton decreases with the increasing angles of  in between 45 ° and 90 °.In this investigation, the above results also imply that slowly and steadily the phase velocity of the DA soliton will increase but later, it will increase rapidly with the increasing values of  in the plasma.When the parameters are supposed to be constant except , the phase velocity of the DA soliton rapidly increases with the rapid increase of critical value of .The above results also indicate that with the increase in the magnitude of external magnetic field in DASWs, the width of the soliton decreases adversely.The results obtained in this problem may be useful for understanding the propagation of Characteristics of Nonlinear Dust Acoustic Waves (DAWS) Propagating... EEJP. 1 (2024)  =  −   ( + 1)(3 −  ) +  ( + 1)(3 −  ) − ,
, we have observed that the width of DASWs vary adversely with the increasing or decreasing values of number density of relative electrons ( ) depending on the various parametric values of nonthermal parameter ().The above figures also indicate that with the increase in Characteristics of Nonlinear Dust Acoustic Waves (DAWS) Propagating... EEJP. 1 (2024)