Ion-acoustic shock waves in magnetized pair-ion plasma
T. Yeashna, R.K. Shikha, N.A. Chowdhury, A. Mannan, S. Sultana, A.A. Mamun
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Ion-acoustic shock waves in magnetized pair-ion plasma
T. Yeashna ∗ , , R.K. Shikha ∗∗ , , N.A. Chowdhury ∗∗∗ , , A. Mannan † , , , S. Sultana ‡ , , and A.A. Mamun § , Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh Institut f¨ur Mathematik, Martin Luther Universit¨at Halle-Wittenberg, Halle, Germanye-mail: ∗ [email protected], ∗∗ [email protected], ∗∗∗ [email protected], † [email protected], ‡ [email protected], § mamun [email protected] Abstract
A theoretical investigation associated with obliquely propagating ion-acoustic shock waves (IASHWs) in a three-component mag-netized plasma having inertialess non-extensive electrons, inertial warm positive and negative ions has been performed. A Burgersequation is derived by employing the reductive perturbation method. Our plasma model supports both positive and negative shockstructures under the consideration of non-extensive electrons. It is found that the positive and negative shock wave potentials in-crease with the oblique angle ( δ ) which arises due to the external magnetic field. It is also observed that the magnitude of theamplitude of positive and negative shock waves is not e ff ected by the variation of the ion kinematic viscosity but the steepnessof the positive and negative shock waves decreases with ion kinematic viscosity. The implications of our findings in space andlaboratory plasmas are briefly discussed. Keywords:
Pair-ion, Magnetized plasma, Ion-acoustic waves, Perturbation methods, Shock waves.
1. Introduction
The pair-ion (PI) plasma can be observed in astrophysi-cal environments such as upper regions of Titan’s atmosphere[1, 2, 3, 4, 5, 6, 7, 8], cometary comae [9], ( H + , O − ) and( H + , H − ) plasmas in the D and F-regions of Earth’s ionosphere[2, 3, 4, 5, 6, 7], and also in the laboratory experiments namely,( Ar + , F − ) plasma [10], ( K + , S F − ) plasma [11, 12], neutralbeam sources [13], plasma processing reactors [14], ( Ar + , S F − )plasma [15, 16, 17, 18], combustion products [19], plasmaetching [19], ( Xe + , F − ) plasma [20], ( Ar + , O − ) plasma, andFullerene ( C + , C − ) plasma [21, 22, 23], etc. Positive ions areproduced by electron impact ionization, and negative ions areproduced by attachment of the low energy electrons. A num-ber of authors studied the nonlinear electrostatic structures inPI plasma [3, 4, 5, 6, 7, 8].Highly energetic particles have been observed in the galaxyclusters [24], the Earth’s bow-shock [25], in the upper iono-sphere of Mars [26], in the vicinity of the Moon [27], and inthe magnetospheres of Jupiter and Saturn [28]. Maxwellianvelocity distribution demonstrating the thermally equilibriumstate of particles is not appropriate for explaining the dynam-ics of these highly energetic particles. Renyi [29] first intro-duced the non-extensive q -distribution for explaining the dy-namics of these highly energetic particles, and further develop-ment of q -distribution has been demonstrated by Tsallis [30].The parameter q in the non-extensive q -distribution describesthe deviation of the plasma particles from the thermally equilib-rium state. It should be noted that q = q < q >
1) refers to super-extensivity (sub-extensivity).Jannat et al. [7] investigated the ion-acoustic (IA) shock waves (IASHWs) in PI plasma in the presence of non-extensive elec-trons, and observed that the height of the positive potential de-creases (increases) with positive (negative) ion mass. Hussain et al. [31] considered inertial PI and inertialess non-extensiveelectrons and investigated IASHWs by considering kinematicviscosities of both positive and negative ion species, and ob-served that the amplitude of the positive IASHWs decreaseswith q . Tribeche et al. [32] studied IA solitary waves in atwo-component plasma, and found that the magnitude of theamplitude of positive and negative solitary structures increaseswith super-extensive and sub-extensive electrons.A plasma medium having considerable dissipative propertiesdictates the formation of shock structures [33, 34, 35]. The Lan-dau damping, kinematic viscosity among the plasma species,and the collision between plasma species are the major causesof the dissipation which is mainly responsible for the formationof shock structures in the plasma medium [33, 34, 35]. Thepresence of kinematic viscosity plays a pivotal role in gener-ating nonlinear waves [33, 34, 35]. Hafez et al. [33] observedthat the steepness of the IASHWs decreases with the increase ofion kinematic viscosity but the amplitude of IASHWs remainsunchanged. Abdelwahed et al. [34] investigated IASHWs in PIplasma and reported that the kinematic viscosity coe ffi cient ofthe ion reduces the steepness of the IASHWs.The external magnetic field is to be considered to change thedynamics of the plasma medium, and associated electrostaticnonlinear structures. Hossen et al. [35] studied the electrostaticshock structures in magnetized dusty plasma, and found that themagnitude of the positive and negative shock profiles increaseswith the oblique angle ( δ ) which arises due to the external mag- Preprint submitted to “The European Physical Journal D” February 17, 2021 etic field. El-Labany et al. [8] considered a three-componentplasma model having inertial PI and inertialess non-extensiveelectrons, and investigated IASHWs, and found that the am-plitude of the positive shock profile decreases with q . To thebest knowledge of the authors, no attempt has been made tostudy the IASHWs in a three-component magnetized plasma byconsidering kinematic viscosities of both inertial warm positiveand negative ion species, and inertialess non-extensive elec-trons. The aim of the present investigation is, therefore, toderive Burgers’ equation and investigate IASHWs in a three-component magnetized PI plasma, and to observe the e ff ects ofvarious plasma parameters on the configuration of IASHWs.The outline of the paper is as follows: The basic equationsare displayed in section 2. The Burgers equation has been de-rived in section 3. Results and discussion are reported in section4. A brief conclusion is provided in section 5.
2. Governing equations
We consider a magnetized plasma system comprising iner-tial negatively and positively charged warm ions, and inertia-less electrons featuring q -distribution. An external magneticfield B has been considered in the system directed along the z -axis defining B = B ˆ z , where B and ˆ z are the strengthof the external magnetic field and unit vector directed alongthe z -axis, respectively. The dynamics of the magnetized PIplasma system is governed by the following set of equations[36, 37, 38, 39, 40, 41, 42] ∂ ˜ n + ∂ ˜ t + ´ ∇ · (˜ n + ˜ u + ) = , (1) ∂ ˜ u + ∂ ˜ t + ( ˜ u + · ´ ∇ ) ˜ u + = − Z + em + ´ ∇ ˜ ψ + Z + eB m + ( ˜ u + × ˆ z ) − m + n + ´ ∇ P + + ˜ η + ´ ∇ ˜ u + , (2) ∂ ˜ n − ∂ ˜ t + ´ ∇ · (˜ n − ˜ u − ) = , (3) ∂ ˜ u − ∂ ˜ t + ( ˜ u − · ´ ∇ ) ˜ u − = Z − em − ´ ∇ ˜ ψ − Z − eB m − ( ˜ u − × ˆ z ) − m − ˜ n − ´ ∇ P − + ˜ η − ´ ∇ ˜ u − , (4)´ ∇ ˜ ψ = π e [˜ n e + Z − ˜ n − − Z + ˜ n + ] , (5)where ˜ n + (˜ n − ) is the positive (negative) ion number density, m + ( m − ) is the positive (negative) ion mass, Z + ( Z − ) is the chargestate of the positive (negative) ion, e being the magnitude ofelectron charge, ˜ u + ( ˜ u − ) is the positive (negative) ion fluid ve-locity, ˜ η + ( ˜ η − ) is the kinematic viscosity of the positive (neg-ative) ion, P + ( P − ) is the pressure of positive (negative) ion,and ˜ ψ represents the electrostatic wave potential. Now, weare introducing normalized variables, namely, n + → ˜ n + / n + , n − → ˜ n − / n − , and n e → ˜ n e / n e , where n − , n + , and n e arethe equilibrium number densities of the negative ions, positiveions, and electrons, respectively; u + → ˜ u + / C − , u − → ˜ u − / C − [where C − = ( Z − k B T e / m − ) / , k B being the Boltzmann con-stant, and T e being temperature of the electron]; ψ → ˜ ψ e / k B T e ; t = ˜ t /ω − P − [where ω − P − = ( m − / π e Z − n − ) / ]; ∇ = ´ ∇ /λ D [where λ D = ( k B T e / π e Z − n − ) / ]. The pressure term of the positiveand negative ions can be recognized as P ± = P ± ( N ± / n ± ) γ with P ± = n ± k B T ± being the equilibrium pressure of thepositive (for + − T + ( T − ) being the temperature of warm positive (negative) ion, and γ = ( N + / N (where N is the degree of freedom and for three-dimensional case N =
3, then γ = / η + ≈ ˜ η − = η ), and η is normalized by ω p − λ D . Thequasi-neutrality condition at equilibrium for our plasma modelcan be written as n e + Z − n − ≈ Z + n + . Equations (1) − (5) canbe expressed in the normalized form as [7, 8]: ∂ n + ∂ t + ∇ · ( n + u + ) = , (6) ∂ u + ∂ t + ( u + · ∇ ) u + = − α ∇ ψ + α Ω c ( u + × ˆ z ) − α ∇ n γ − + + η ∇ u + , (7) ∂ n − ∂ t + ∇ · ( n − u − ) = , (8) ∂ u − ∂ t + ( u − · ∇ ) u − = ∇ ψ − Ω c ( u − × ˆ z ) − α ∇ n γ − − + η ∇ u − , (9) ∇ ψ = µ e n e − (1 + µ e ) n + + n − . (10)Other plasma parameters are defined as α = Z + m − / Z − m + , α = γ T + m − / ( γ − Z − T e m + , α = γ T − / ( γ − Z − T e , µ e = n e / Z − n − , and Ω c = ω c /ω p − [where ω c = Z − eB / m − ]. Now,the expression for the number density of electrons followingnon-extensive q -distribution can be written as [8] n e = h + ( q − ψ i q + q − , (11)where the parameter q represents the non-extensive propertiesof electrons. We have neglected the e ff ect of the external mag-netic field on the non-extensive electron distribution. This isvalid due to the fact that the Larmor radii of electrons is so smallthat as if the electrons are flowing along the magnetic field linesof force. Now, by substituting Eq. (11) into the Eq. (10), andexpanding up to third order in ψ , we get ∇ ψ = µ e + n − − (1 + µ e ) n + + σ ψ + σ ψ + σ ψ + · · · , (12)where σ = [ µ e ( q + / , σ = [ µ e ( q + − q )] / ,σ = [ µ e ( q + − q )(5 − q )] / . We note that the terms containing σ , σ , and σ are the con-tribution of q -distributed electrons.
3. Derivation of the Burgers’ equation
To derive the Burgers’ equation for the IASHWs propagatingin a magnetized PI plasma, first we introduce the stretched co-ordinates [35, 43] ξ = ǫ ( l x x + l y y + l z z − v p t ) , (13) τ = ǫ t , (14)2here v p is the phase speed and ǫ is a smallness parameter mea-suring the weakness of the dissipation (0 < ǫ < l x , l y ,and l z (i.e., l x + l y + l z =
1) are the directional cosines of thewave vector k along x , y , and z -axes, respectively. Then, thedependent variables can be expressed in power series of ǫ as[35] n + = + ǫ n (1) + + ǫ n (2) + + ǫ n (3) + + · · · , (15) n − = + ǫ n (1) − + ǫ n (2) − + ǫ n (3) − + · · · , (16) u + x , y = ǫ u (1) + x , y + ǫ u (2) + x , y + · · · , (17) u − x , y = ǫ u (1) − x , y + ǫ u (2) − x , y + · · · , (18) u + z = ǫ u (1) + z + ǫ u (2) + z + · · · , (19) u − z = ǫ u (1) − z + ǫ u (2) − z + · · · , (20) ψ = ǫψ (1) + ǫ ψ (2) + · · · . (21)Now, by substituting Eqs. (13) − (21) into Eqs. (6) − (9), and(12), and collecting the terms containing ǫ , the first-order equa-tions reduce to n (1) + = α l z v p − α l z ψ (1) , (22) u (1) + z = v p α l z v p − α l z ψ (1) , (23) n (1) − = − l z v p − α l z ψ (1) , (24) u (1) − z = − v p l z v p − α l z ψ (1) . (25)Now, the phase speed of IASHWs can be written as v p ≡ v p + = l z vut − a + q a − σ a σ , (26) v p ≡ v p − = l z vut − a − q a − σ a σ , (27)where a = − − α σ − α σ − α µ e − α and a = α + α α σ + α α µ e + α α . The x and y -components ofthe first-order momentum equations can be manifested as u (1) + x = − l y v p Ω c (3 v p − α l z ) ∂ψ (1) ∂ξ , (28) u (1) + y = l x v p Ω c (3 v p − α l z ) ∂ψ (1) ∂ξ , (29) u (1) − x = − l y v p Ω c (3 v p − α l z ) ∂ψ (1) ∂ξ , (30) u (1) − y = l x v p Ω c (3 v p − σ l z ) ∂ψ (1) ∂ξ . (31)Now, by taking the next higher-order terms, the equation ofcontinuity, momentum equation, and Poisson’s equation can be μ e - - - A q - - A Figure 1: The variation of nonlinear coe ffi cient A with µ e when q = . ffi cient A with q when µ e = . α = . α = . α = . δ = ◦ ,and v p ≡ v p + . written as ∂ n (1) + ∂τ − v p ∂ n (2) + ∂ξ + l x ∂ u (1) + x ∂ξ + l y ∂ u (1) + y ∂ξ + l z ∂ u (2) + z ∂ξ + l z ∂∂ξ (cid:0) n (1) + u (1) + z (cid:1) = , (32) ∂ u (1) + z ∂τ − v p ∂ u (2) + z ∂ξ + l z u (1) + z ∂ u (1) + z ∂ξ + α l z ∂ψ (2) ∂ξ + α l z ∂∂ξ (cid:20) n (2) + −
19 ( n (1) + ) (cid:21) − η ∂ u (1) + z ∂ξ = , (33) ∂ n (1) − ∂τ − v p ∂ n (2) − ∂ξ + l x ∂ u (1) − x ∂ξ + l y ∂ u (1) − y ∂ξ + l z ∂ u (2) − z ∂ξ + l z ∂∂ξ (cid:0) n (1) − u (1) − z (cid:1) = , (34) ∂ u (1) − z ∂τ − v p ∂ u (2) − z ∂ξ + l z u (1) − z ∂ u (1) − z ∂ξ − l z ∂ψ (2) ∂ξ + α l z ∂∂ξ (cid:20) n (2) − −
19 ( n (1) − ) (cid:21) − η ∂ u (1) − z ∂ξ = , (35) σ ψ (2) + σ [ ψ (1) ] + n (2) − − ( µ e + n (2) + = . (36)Finally, the next higher-order terms of Eqs. (6) − (9), and (12),with the help of Eqs. (22) − (36), can provide the Burgers equa-tion as ∂ Ψ ∂τ + A Ψ ∂ Ψ ∂ξ = C ∂ Ψ ∂ξ , (37)where Ψ = ψ (1) is used for simplicity. In Eq. (37), the nonlinearcoe ffi cient A and dissipative coe ffi cient C are given by A = α v p s l z + F v p s l z s + F , and C = η , (38)where F = µ e α v p s l z − v p s l z + µ e α α s l z + α α s l z + α s l z − σ s s , F = α v p s l z s + α µ e v p s l z s , s = v p − α l z , s = v p − α l z . Now, we look for stationary shock wave solution of this Burg-ers’ equation by considering ζ = ξ − U τ ′ and τ = τ ′ (where U is the speed of the shock waves in the reference frame).3 = °δ (cid:0)3(cid:1) °δ (cid:2) °- -
100 100 200 ζ ψ Figure 2: The variation of Ψ with ζ for di ff erent values of δ under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . α = . η = . µ e = . q = . U = .
01, and v p ≡ v p + . δ = ° δ = (cid:6)(cid:7) ° δ = °- -
100 100 200 ζ - (cid:8)(cid:9)(cid:10)(cid:11) - - (cid:12)(cid:13)(cid:14)(cid:15) ψ Figure 3: The variation of Ψ with ζ for di ff erent values of δ under the consid-eration µ e < µ ec . Other plasma parameters are α = . α = . α = . η = . µ e = . q = . U = .
01, and v p ≡ v p + . These allow us to write the stationary shock wave solution as[35, 44, 45]
Ψ = Ψ m h − tanh (cid:18) ζ ∆ (cid:19)i , (39)where the amplitude Ψ m and width ∆ are given by Ψ m = U A , and ∆ = CU . (40)It is clear from Eqs. (39) and (40) that the IASHWs exist, whichare formed due to the balance between nonlinearity and dissi-pation, because C > Ψ > Ψ < A > A <
0) because U >
4. Results and discussion
The balance between nonlinearity and dissipation leads togenerate IASHWs in a three-component magnetized PI plasma.We have numerically analyzed the variation of A with µ e in theleft panel of Fig. 1, and it is obvious from this figure that (a) A can be negative, zero, and positive depending on the values of µ e ; (b) the value of µ e for which A becomes zero is known ascritical value of µ e (i.e., µ ec ), and the µ ec for our present analysisis almost 0 .
2; and (c) the parametric regimes for the formationof positive (i.e., ψ >
0) and negative (i.e., ψ <
0) potential shockstructures can be found corresponding to A > A <
0. The η = (cid:16)(cid:17)(cid:18) η = (cid:19)(cid:20)(cid:21) η = (cid:22)(cid:23)(cid:24) - -
100 100 200 ζ (cid:25)(cid:26)(cid:27)(cid:28) ψ Figure 4: The variation of Ψ with ζ for di ff erent values of η under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . α = . δ = ◦ , η = . µ e = . q = . U = .
01, and v p ≡ v p + . η = (cid:29)(cid:30)(cid:31) η = !" η = - -
100 100 200 ζ - - - - &’() - - - ψ Figure 5: The variation of Ψ with ζ for di ff erent values of η under the consid-eration µ e < µ ec . Other plasma parameters are α = . α = . α = . δ = ◦ , η = . µ e = . q = . U = .
01, and v p ≡ v p + . right panel of Fig. 1 describes the variation of A with q whenother plasma parameters are constant and in this case, A be-comes zero for the critical value of q (i.e., q = q c ≃ . q > . q < . µ e > µ ec and negativepotential shock structure under the consideration µ e < µ ec withthe oblique angle ( δ ), respectively. It is clear from these figuresthat (a) the magnitude of the amplitude of positive and negativepotential structures increases with an increase in the value of the δ , and this result agrees with the result of Hossen et al. [35];(b) the magnitude of the negative potential is always greaterthan the positive potential for same plasma parameters. So, theoblique angle enhances the amplitude of the potential profiles.Figures 4 and 5 illustrate the e ff ects of the ion kinematic vis-cosity on the positive (under the consideration µ e > µ ec ) andnegative (under the consideration µ e < µ ec ) shock profiles. It isreally interesting that the magnitude of the amplitude of posi-tive and negative shock profiles is not e ff ected by the variationof the ion kinematic viscosity but the steepness of the shockprofile decreases with ion kinematic viscosity, and this resultagrees with the previous work of Refs. [33, 34].The e ff ects of the sub-extensive electrons (i.e., q >
1) on thepositive potential profile can be seen in Fig. 6 under the consid-eration µ e > µ ec . The height of the positive potential decreases4 = = = - -
100 100 200 ζ ,-./ ψ Figure 6: The variation of Ψ with ζ for di ff erent values of q under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . α = . δ = ◦ , η = . µ e = . U = .
01, and v p ≡ v p + . q =- q =- =- - -
100 100 200 ζ - - - - ψ Figure 7: The variation of Ψ with ζ for di ff erent values of q under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . α = . δ = ◦ , η = . µ e = . U = .
01, and v p ≡ v p + . q = = = - -
100 100 200 ζ - - - - ψ Figure 8: The variation of Ψ with ζ for di ff erent values of q under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . α = . δ = ◦ , η = . µ e = . U = .
01, and v p ≡ v p + . with q , and this result is a good agreement with the result ofEl-Labany et al. [8] and Hussain et al. [31]. Figures 7 and8 illustrate the role of super-extensive electrons (i.e., q < µ e > µ ec , and this is really interesting that the existenceof the super-extensive electron produces negative potential, andthe magnitude of the amplitude of negative potential increaseswith q . So, the orientation of the potential profiles (positive andnegative) has been organized by the sign of q under the consid-eration µ e > µ ec .It can be seen from the literature that the PI plasma sys- α = α = α = - -
100 100 200 ζ ψ Figure 9: The variation of Ψ with ζ for di ff erent values of α under the consid-eration µ e > µ ec . Other plasma parameters are α = . α = . δ = ◦ , η = . µ e = . q = . U = .
01, and v p ≡ v p + . α = α = α = - -
100 100 200 ζ - - - - ψ Figure 10: The variation of Ψ with ζ for di ff erent values of α under the con-sideration µ e > µ ec . Other plasma parameters are α = . α = . δ = ◦ , η = . µ e = . q = . U = .
01, and v p ≡ v p + . tem can support these conditions: m − > m + (i.e., H + − O − [2, 3, 4, 5, 6, 7], Ar + − S F − [15, 16, 17, 18], and Xe + − S F − [15, 16, 17, 18]), m − = m + (i.e., H + − H − [2, 3, 4, 5, 6, 7] and C + − C − [21, 22, 23]), and m − < m + (i.e., Ar + − F − [3, 4]).So, in our present investigation, we have graphically observedthe variation of the electrostatic positive potential with α un-der the consideration of m − > m + (i.e., α >
1) and µ e > µ ec inFig. 9, and it is obvious from this figure that (a) the amplitudeof the positive potential decreases with an increase in the valueof the negative ion mass but increases with an increase in thevalue of the positive ion mass for a fixed value of their chargestate; (b) the height of the IASHWs with positive potential in-creases (decreases) with negative (positive) ion charge state fora constant mass of positive and negative ion species. So, themass and charge state of the PI play an opposite role for theformation of positive shock structure. Figure 10 describes thenature of the electrostatic negative potential with α under theconsideration of m − < m + (i.e., α <
1) and µ e > µ ec . It is clearfrom this figure that (a) due to the m − < m + (i.e., α < µ e > µ ec (i.e., A > − > m + (i.e., α >
1) and m − < m + (i.e., α <
5. Conclusion
We have studied IASHWs in a three-component magnetizedPI plasma by considering kinematic viscosities of both inertialwarm positive and negative ion species, and inertialess non-extensive electrons. The reductive perturbation method [46] isused to derive the Burgers’ equation. The results that have beenfound from our investigation can be summarized as follows: • The parametric regimes for the formation of positive (i.e., ψ >
0) and negative (i.e., ψ <
0) potential shock structurescan be found corresponding to A > A < • The magnitude of the amplitude of positive and negativeshock structures increases with the oblique angle ( δ ) whicharises due to the external magnetic field. • The magnitude of the amplitude of positive and negativeshock profiles is not e ff ected by the variation of the ionkinematic viscosity but the steepness of the shock profiledecreases with ion kinematic viscosity.It may be noted here that the gravitational e ff ect is very impor-tant but beyond the scope of our present work. In future andfor better understanding, someone can investigate the nonlin-ear propagation in a three-component PI plasma by consider-ing the gravitational e ff ect. The results of our present investi-gation will be useful in understanding the nonlinear phenom-ena both in astrophysical environments such as upper regionsof Titan’s atmosphere [1, 2, 3, 4, 5, 6, 7, 8], cometary comae[9], ( H + , O − ) and ( H + , H − ) plasmas in the D and F-regions ofEarth’s ionosphere [2, 3, 4, 5, 6, 7], and also in the laboratoryexperiments, namely, ( Ar + , F − ) plasma [10], ( K + , S F − ) plasma[11, 12], neutral beam sources [13], plasma processing reactors[14], ( Ar + , S F − ) plasma [15, 16, 17, 18], combustion products[19], plasma etching [19], ( Xe + , F − ) plasma [20], ( Ar + , O − )plasma, and Fullerene ( C + , C − ) plasma [21, 22, 23], etc. References [1] A.J. Coates, et al. , Geophys. Res. Lett. , L22103 (2007).[2] H. Massey, Negative Ions , 3rd ed., (Cambridge University Press, Cam-bridge, 1976).[3] R. Sabry, et al. , Phys. Plasmas , 032302 (2009).[4] H.G. Abdelwahed, et al. , Phys. Plasmas , 022102 (2016).[5] A. P. Misra, Phys. Plasmas, , 033702 (2009).[6] A. Mushtaq, et al. , Phys. Plasmas , 042304 (2012).[7] N. Jannat, et al. , Commun. Theor. Phys. , 479 (2015).[8] S.K. El-Labany, et al. , Eur. Phys. J. D , 104 (2020); N.A. Chowdhury, et al. , Chaos , 093105 (2017); N. Ahmed, et al. , Chaos , 123107(2018); M. Hassan, et al. , Commun. Theor. Phys. , 1017 (2019); S.Jahan, et al. , Plasma Phys. Rep. , 90 (2020).[9] P.H. Chaizy, et al. , Nature (London), , 393 (1991).[10] Y. Nakamura, I. Tsukabayashi, Phys. Rev. Lett. , 2356 (1984).[11] B. Song, et al. , Phys. Fluids B , 284 (1991).[12] N. Sato, Plasma Sources Sci. Technol. , 395 (1994).[13] M. Bacal, G.W. Hamilton, Phys. Rev. Lett. , 1538 (1979).[14] R.A. Gottscho, C.E. Gaebe, IEEE Trans. Plasma Sci. , 92 (1986).[15] A.Y. Wong, et al. , Phys. Fluids , 1489 (1975).[16] Y. Nakamura, et al. , Plasma Phys. Control. Fusion , 105 (1997). [17] J.L. Cooney, et al. , Phys. Fluids B , 2758 (1991).[18] Y. Nakamura, et al. , Phys. Plasmas , 3466 (1999).[19] D.P. Sheehan, N. Rynn, Rev. Sci. lnstrum. , 8 (1988).[20] R. Ichiki, et al. , Phys. Plasmas , 4481 (2002).[21] W. Oohara, R. Hatakeyama, Phys. Rev. Lett. , 205005 (2003).[22] R. Hatakeyama, W. Oohara, Phys. Scripta , 101 (2005).[23] W. Oohara, et al. , Phys. Rev. Lett. , 175003 (2005).[24] S.H. Hansen, New Astron. , 371 (2005).[25] J.R. Asbridge, et al. , J. Geophys. Res. , 5777 (1968).[26] R. Lundlin, et al. , Nature (London) , 609 (1989).[27] Y. Futaana, et al. , J. Geophys. Res. , 1025 (2003).[28] S.M. Krimigis, et al. , J. Geophys. Res. , 8871 (1983).[29] A. R´enyi, Acta Math. Acad. Sci. Hung. , 285 (1955).[30] C. Tsallis, J. Stat. Phys. , 479 (1988).[31] S. Hussain, et al. , Phys. Plasmas , 092303 (2013).[32] M. Tribeche, L. Djebarni, R. Amour, Phys. Plasmas , 042114 (2010).[33] M.G. Hafez, et al. , Plasma Phys. Rep. , 499 (2017).[34] H.G. Abdelwahed, et al. , J. Exp. Theor. Phys. , 1111 (2016).[35] M.M. Hossen, et al. , High Energy Density Phys. , 9 (2017).[36] A. Atteya, S. Sultana, R. Schlickeiser, Chin. J. Phys. , 1931 (2018).[37] N.C. Adhikary, Phys. Lett. A , 1460 (2012).[38] A.N. Dev, M.K. Deka, Phys. Plasmas , 072117 (2018).[39] A.N. Dev, et al. , Chin. Phys. B , 105202 (2016).[40] A.N. Dev, et al. , Commun. Theor. Phys. , 875 (2014).[41] M.K. Deka, A.N. Dev, Plasma Phys. Rep. , 965 (2018).[42] B. Sahu, A. Sinha, R. Roychoudhury, Phys. Plasmas , 103701 (2014).[43] H. Washimi, T. Taniuti, Phys. Rev. Lett. , 996 (1966).[44] V.I. Karpman, Nonlinear Waves in Dispersive Media , (Pergamon Press,Oxford, 1975).[45] A. Hasegawa,
Plasma Instabilities and Nonlinear E ff ects , (Springer-Verlag, Berlin, 1975).[46] M.H. Rahman, et al. , Phys. Plasmas , 102118 (2018); N.A. Chowdhury, et al. , Phys. plasmas , 113701 (2017); M.H. Rahman, et al. , Chin. J.Phys. , 2061 (2018); N.A. Chowdhury, et al. , Vacuum , 31 (2018);R.K. Shikha, et al. , Eur. Phys. J. D , 177 (2019); N.A. Chowdhury, et al. , Contrib. Plasma Phys. , 870 (2018); N.A. Chowdhury, et al. ,Plasma Phys. Rep. , 459 (2019); S.K. Paul, et al. , Pramana J. Phys ,58 (2020); T.I. Rajib, et al. , Phys. plasmas , 123701 (2019); S. Jahan, et al. , Commun. Theor. Phys. , 327 (2019)., 327 (2019).