Electrostatic Nonplanar Positron-Acoustic Shock Waves in Superthermal Electron-Positron-Ion Plasmas

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1 Commun. Theor. Phys ) Vol. 63, No. 6, June 1, 2015 Electrostatic Nonplanar Positron-Acoustic Shock Waves in Superthermal Electron-Positron-Ion Plasmas M.J. Uddin, 1, M.S. Alam, 2 and A.A. Mamun 3 Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Received December 15, 2014; revised manuscript received March 30, 2015) Abstract The basic properties of the nonlinear propagation of the nonplanar cylindrical and spherical) positronacoustic PA) shock waves SHWs) in an unmagnetized electron-positron-ion e-p-i) plasma containing immobile positive ions, mobile cold positrons, and superthermal kappa distributed) hot positrons and electrons are investigated both analytically and numerically. The modified Burgers equation mbe) is derived by using the reductive perturbation method. The basic features of PA SHWs are significantly modified by the cold positron kinematic viscosity η), superthermal parameter of electrons κ e), superthermal parameter of hot positrons κ p), the ratio of the electron temperature to hot positron temperature σ), the ratio of the electron number density to cold positron number density µ e), and the ratio of the hot positron number density to cold positron number density µ ph ). This study could be useful to identify the basic properties of nonlinear electrostatic disturbances in dissipative space and laboratory plasmas. PACS numbers: Kn, h, Ep, Mw, Tc, j Key words: positron-acoustic waves, superthermal particles, Kappa distribution, shock waves, modified Burgers equation, electron-positron-ion plasmas josim.phys2007@gmail.com c 2015 Chinese Physical Society and IOP Publishing Ltd 1 Introduction It is well known that electron-positron-ion e-p-i) plasma is a fully ionized gas comprising of electrons and positrons having equal masses and charges with opposite polarity. The e-p-i plasmas are found not only in early universe [1 4] but also in astrophysical environments such as in neutron stars and intense laser solid density plasma experiments, [5] dense white dwarfs, [6] pulsar magnetospheres, [7 8] polar coups region of pulsars and around active galactic nuclie, [9] certain heliospheric environments, [10 11] ionosphere and the auroral acceleration regions, [12] interstellar medium, [13] etc. Besides, e-p-i plasma can be artificially produced in laboratories. [14 15] The study on the nonlinear propagation of shock waves in electron-positron-ion e-p-i) plasmas has drawn a significant attention in recent decades. It is well established that the behavior of the linear and nonlinear waves in e-p-i plasmas notably changed due to the presence of positrons in such electron-ion plasmas. Masood et al. [16] investigated the ion-acoustic shock waves IASHWs) in an unmagnetized plasma consisting of electrons, positrons and adiabatically hot positive ions. They observed that the positron concentration, ratio of ion to electron temperature, and the plasma kinematic viscosity significantly modifies the shock structures. One year later, Masood et al. [17] studied the IASHWs with weak transverse perturbations in an unmagnetized plasma consisting of electrons, positrons and singly charged adiabatically hot positive ions. In the same year Masood and Rizvi [18] analyzed the two-dimensional planar and nonplanar IASHWs by considering the same plasma model. Asif Shah et al. [19] studied the nonplanar converging and diverging shock waves in e-p-i plasmas containing inertialess electrons, positrons, and inertial thermal ions. They found that the strength of shock in spherical geometry dominates over the shock strength in cylindrical geometry. Chatterjee et al. [20] investigated the nonlinear propagation of IASHWs in an unmagnetized plasma consisting of nonthermal electrons, nonthermal positrons, and singly charged adiabatically hot positive ions, whose dynamics is governed by the twodimensional nonplanar Kadomstev Petviashvili Burgers KPB) equation. Very recently, Ata-ur-Rahman et al. [21] presented the planar and nonplanar IASHWs in an unmagnetized and dissipative plasma system comprising of non-degenerate cold ions, relativistic degenerate electrons, and positrons. They derived the Korteweg-de Vries Burgers KdVB) and modified KdVB equations by using the reductive perturbation technique and analyzed both analytically and numerically. Energetic particles viz. electrons, positrons, and ions are found in various astrophysical plasmas. All of this high energy particles superthermal particles) originates because of the results of the interaction of wave particles or due to the effects of external forces on the space plasma environments. A high energy long tail appears in the distribution function of such plasma particles. Maxwell Boltzmann distribution is inadequate to model the energetic space plasma particles. The kappa κ) distribution was first suggested by Vasyliunas [22] to model such space plasmas. Kappa distribution function

2 No. 6 Communications in Theoretical Physics 755 in three-dimensional form can be scripted in the following form: [22 30] F κ v) = Γκ + 1) ) 1 + v2 κ+1), πκθ 2 ) 3/2 Γκ 1/2) κθ 2 where Γ is the gamma function, θ is the effective thermal speed or most probable speed connected to the usual thermal velocity V t = k B T/m) 1/2 by θ = [2κ 3)/κ]V t, T is the characteristic kinetic temperature, and k B is the Boltzmann constant. The parameter κ represents the strength of superthermality of the plasma species. The range of this parameter is 3/2 < κ <. The advantage of using κ distribution lies in the fact that the Maxwellian distribution is a special case of the κ function in the limit of κ. During the last few decades, many theoretical investigations have been made on e-p-i plasmas [31] with superthermal kappa distributed) particles to study the nonlinear shock waves. Pakzad [32] studied the effects of Boltzmann distributed positrons and superthermal electrons on the nonlinear propagation of IASHWs in e-p-i plasmas. He observed that the amplitude of the IASHWs decreases with the increase of positron concentration. El- Bedwehy and Moslem [33] investigated the propagation of three-dimensional ion-acoustic solitary and shock waves in e-p-i magnetoplasma with superthermal electrons and positrons represented by kappa distribution). Shah and Saeed [34] studied the IASHWs in e-p-i plasmas whose components are relativistic adiabatic ions, kappa distributed electrons, and positrons. Sahu [35] studied the ion-acoustic shock and solitary structures in an unmagnetized e-p-i plasma consisting of nonextensive electrons and thermal positrons in bounded nonplanar geometry. The IASHWs in e-p-i plasmas comprising of relativistic warm ions, thermal positrons and nonextensive electrons investigated by Tribeche and Pakzad. [36] They observed that the strength and steepness of shock waves significantly modified by the relativistic, nonextensive, and dissipative effects. Pakzad and Tribeche [37] studied the nonlinear propagation of IASHWs in dissipative e-p-i plasmas with superthermal electrons, Maxwell Boltzmann distributed positrons and relativistic ions. The monotonic and oscillatory IASHWs in e-p-i plasmas consisting of adiabatically heated warm) ions, nonthermal kappa distributed electrons and positrons studied by Hussain et al. [38] Masud et al. [39] studied the dust-ion-acoustic DIA) shock waves in an unmagnetized e-p-i plasmas consisting of inertial ions, two temperature low and high) superthermal electrons, Boltzmann distributed positrons and negatively charged immobile dust grains and discussed the basic feature of DIA shock waves. The spherical and cylindrical IASHWs in e-p-i plasmas consisting of cold ions, nonextensive electrons and thermal positrons analyzed by Eslami et al. [40] Alam et al. [29] studied the properties of dust-ionacoustic DIA) shock waves in an unmagnetized dusty plasma containing inertial ions, kappa distributed electrons of two distinct temperatures, and negatively charged immobile dust grains). They derived the Burgers equation BE) using the reductive perturbation method. However, their work [29] is limited to a planar geometry, which is not appropriate for many space and laboratory plasma systems of spherical or cylindrical shape e.g. capsule implosion spherical geometry), shock tube cylindrical geometry), star formation, and supernova explosions, etc.). Therefore, in our present work we consider nonplanar cylindrical and spherical) geometry to examine the properties of positron-acoustic PA) shock waves in a four component electron-positron-ion plasma system consisting of immobile positive ions, mobile cold positrons, and superthermal kappa distributed) positrons and electrons. Our present work differs from the work of Alam et al. [29] in some other points that i) They considered kappa distributed electrons of two distinct temperatures, but we have considered in our present work superthermal kappa distributed) positrons and electrons; ii) They considered negatively charged immobile dust grains, but we considered here immobile positive ions; iii) The source of dissipation in their work is ion-kinematic viscosity, but that in our work is cold positron viscosity. A number of authors investigated the nonlinear propagation of positron-acoustic PA) waves viz. solitary, shocks, double layers, etc.) in e-p-i plasmas. [41 46] In such PA waves inertia comes from the cold positron mass whereas restoring force comes from the essential thermal pressure of hot positrons and electrons. Recently, Sahu [44] studied the nonlinear propagation of PA shock waves in e-p-i plasmas consisting of Boltzmann distributed hot positrons and electrons, mobile cold positrons and immobile ions using the reductive perturbation method. However, his assumption is valid only for isothermal plasmas. Rahman et al. [46] investigated the PA planar shock waves in e-p-i plasmas comprising of nonthermal Cairns) distributed positrons and electrons. Shah et al. [47] studied the PA shock waves in an unmagnetized, collisionless degenerate quantum plasma containing non-relativistic cold positrons, both non-relativistic and ultra-relativistic degenerate electrons, and hot positron fluids and positively charged static ions). All of them did not study the effects of superthermal kappa distributed) electrons and positrons on the PA shock waves. Very recently, Alam et al. [48] worked on PA solitary waves and double layers DLs) in four-component e-p-i plasmas and examined the roles of superthermal electrons and positrons and Uddin et al. [49] studied the PA nonplanar Gardner solitary waves in superthermal e-p-i plasmas. So far we know, no theoretical investigation has been made on the nonplanar cylindrical and spherical) PA shock waves in e-p-i plasmas comprising of superthermal kappa distributed) hot positrons and electrons, cold mobile viscous positrons, and immobile positive ions. There-

3 756 Communications in Theoretical Physics Vol. 63 fore, we have considered an unmagnetized e-p-i plasma system containing immobile positive ions, cold mobile viscous positron fluid, and kappa distributed hot positrons and electrons to study the basic features of nonplanar positron-acoustic shock waves PA SHWs). The layout of this article is as follows. The governing equations are presented in Sec. 2. The modified Burgers equation mbe) is derived in Sec. 3. The numerical results are discussed in Sec. 4. Finally, a brief discussion is provided in Sec Governing Equations We have considered a four component unmagnetized plasma system containing immobile positive ions, cold mobile viscous positrons, and kappa distributed hot positrons and electrons to study the nonlinear propagation of cylindrical and spherical PA SHWs. Hence, at equilibrium n e0 = n 0 + n ph0 + n i0, where n e0, n i0 are the unperturbed electron number density and ion number density respectively. n 0 n ph0 ) is the number density of unperturbed cold hot) positron. The hot electrons and hot positrons are assumed to obey kappa distribution and are given by the following expressions: [50 52] eφ ] κe+1/2 n e = n e0 [1, k B T e κ e 3/2) eφ ] κp+1/2 n ph = n ph0 [1 +, k B T ph κ p 3/2) where n e and n ph are the number densities, while T e and T ph are the temperatures of electrons and hot positrons, respectively. The real parameters κ e and κ p are the superthermal parameters of hot electrons and hot positrons, respectively. The normalized basic equations governing the dynamics of the nonplanar PA SHWs are given in dimensionless variables as follows: n + 1 t r ν r rν n u ) = 0, 1) u u + u = φ t r r + η 1 r ν r ν u ), 2) r r 1 r ν r ν φ ) = ρ, 3) r r ρ = n µ ph 1 + σ 1φ κ p 3/2 + µ e 1 σ 2φ ) κp+1/2 ) κe+1/2 µi, 4) κ e 3/2 where n is the cold positron number density normalized by its equilibrium value n 0, u is the cold positron fluid speed normalized by C = k B T e /m p ) 1/2, k B is the Boltzmann constant, m p is the positron mass, e is the magnitude of the electron charge, σ = T e /T ph, µ ph = n ph0 /n 0, µ e = n e0 /n 0, and µ i = n i0 /n 0, φ is the electrostatic wave potential normalized by k B T e /e, time variable t is normalized by ω 1 = m p /4πn 0 e 2 ) 1/2 and the space variable x is normalized by the Debye length λ D = k B T e /4πn 0 e 2 ) 1/2. It is important to mention here that we are interested in studying the properties of SHWs associated with PA waves. The PA waves are typically high frequency waves in comparison with the ion plasma frequency. On the PA wave time scale, the ions are generally assumed stationary forming a neutralizing background. This means that the ion dynamics does not influence the PA waves because the PA wave frequency is much larger than the ion plasma frequency. Thus, we have considered the cold positron kinematic viscosity. Besides, several authors studied the electron-acoustic EA) SHWs by taking into account the cold electron kinematic viscosity. [53 55] Since, the mass of electron and positron is equal; therefore one can logically consider the cold positron kinematic viscosity for studying the PA SHWs. 3 Derivation of Modified Burgers Equation mbe) We have deduced the modified Burgers equation mbe) to describe the nonlinear dynamics of the PA SHWs. PA mbe has derived to study the solutions of Eqs. 1) 4) by introducing the following stretched coordinates: ζ = ɛr V p t), 5) τ = ɛ 3 t, 6) where V p is the phase speed of the PA SHWs and ɛ is a smallness parameter measuring the weakness of the dispersion 0 < ɛ < 1). To obtain a dynamical equation, we also expand the perturbed quantities n, u, φ, and ρ in power series of ɛ. Let S be any of the system variables n, u, φ, and ρ, describing the systems s state at a given position and instant. We consider small deviations from the equilibrium state S 0) - which explicitly is n 0) = 1, u 0) = 0, φ 0) = 0, and ρ 0) = 0 by taking S = S 0) + ɛ n S n). 7) n=1 To the lowest order in ɛ, Eqs. 1) 7) give u 1) = 1 V p ψ, 8) n 1) = 1 Vp 2 ψ, 9) 1 Vp 2 = µ phκ p 1/2)σ + µ eκ e 1/2), 10) κ p 3/2 κ e 3/2 where ψ = φ 1). Equation 10) represents the dispersion properties of the PA SHWs propagating in a plasma system under consideration. To the next higher order in ɛ, we obtain a set of equations, which, after using Eqs. 7) 10), can be simplified as n 1) τ n 2) V p + u2) + [n1) u 1) ] + νu1) V p τ = 0, 11)

4 No. 6 Communications in Theoretical Physics 757 u 1) τ n 2) u 2) V p + u 1) + µ ph P 1 P 2 σ 2 ψ P 2 3 u 1) + φ2) ψ + µ P 2 σ φ2) ph P 3 η 2 u 1) 2 = 0, 12) P 4 P 5 ψ ψ µ e P6 2 + µ P 5 φ2) e = 0, 13) P 6 where P 1 = 1/2 κ p, P 2 = 1/2 κ p, P 3 = 3/2 + κ p, P 4 = 1/2 κ e, P 5 = 1/2 κ e, and P 6 = 3/2 + κ e. Now, combining Eqs. 11) 13), we obtain a new equation of the form: where ψ τ + Aψ ψ + νψ 2τ = C 2 ψ 2, 14) A = V p 3 [ 3 2 Vp 4 + µ php 1 P 2 σ 2 P3 2 µ ep 4 P ] 5 P6 2, 15) C = η 2, 16) in which P 1 = 1/2 κ p, P 2 = 1/2 κ p, P 3 = 3/2 + κ p, P 4 = 1/2 κ e, P 5 = 1/2 κ e, and P 6 = 3/2 + κ e. Equation 14) is the representation of very well known Burgers equation which is modified by the additional term ν/2τ)ψ which occurs as a result of the effect of the nonplanar cylindrical ν = 1 or spherical ν = 2) geometry. 4 Numerical Results The stationary localized solution of the Burgers equation is given by [56] ψ = ψ m [1 tanhζ/ )], 17) where = 2C/U 0 is the width and ψ m = U 0 /A is the amplitude of the shock waves. It is obvious from Eq. 14), for A > <)0, the plasma system supports compressive rarefactive) PA SHWs which are associated with a positive negative) potential, and no shock waves exist at A = 0. It is obvious that A is a function of µ ph, µ e, κ p, κ e, and σ. Therefore, Aµ ph = µ c ) = 0 and µ c can be expressed as µ ph = µ c = R 1R 2 µ e R 4 R 6 σ R 3 6R 6 R R4 2σ2 + 12R 1 R 2 µ e R 1 R 5 + R 3 R 4 σ), 18) 6R 4 R 6 σ where R 1 = 2κ p 3, R 2 = 2κ e 1, R 3 = 2κ p + 1, R 4 = 2κ e 3, R 5 = 2κ e + 1, R 6 = 2κ p 1. The critical value of µ ph above below) which the shock waves with a positive negative) potential exists is represented by Eq. 18). The critical value of µ ph µ c = µ ph = 0.188) is numerically found for typical plasma parameters, viz. µ e = 0.5, σ = 2, κ p = 3, and κ e = 1.6. It is clear that when µ ph is close to µ c, the amplitude of the shock strictures becomes large enough to break down the validity of the perturbation theory and the mbe derived here is no longer valid. Thus we have choosen the plasma parameters which correspond to µ ph µ c or µ ph µ c. Now we focus on Eq. 14) with the term ν/2τ)ψ, which is due to the effect of the nonplanar cylindrical or spherical) geometry. To the best of our knowledge, since an exact solution of Eq. 14) is not possible to obtain, we have numerically solved Eq. 14), and have studied the consequences of cylindrical ν = 1) and spherical ν = 2) geometries on time-dependent PA SHWs. The results are shown in Figs. 3) 6). The model of the stationary solution of Eq. 14) without the term ν/2τ)ψ is the initial condition, that we have used in our numerical analysis. Fig. 1 Color online) Variation of dependence of nonlinear coefficient A with κ p and κ e. Here µ ph = 0.3, µ e = 0.5, and σ = 2. Fig. 2 Color online) Variation of phase velocity of positive shock profiles with κ p and κ e. Here µ ph = 0.25, µ e = 0.5, and σ = 2. Figures 3 and 4 show the effects of cylindrical geometry on the positive and negative PA SHWs and Figs. 5 and 6 show the effects of spherical geometry on the positive and negative PA SHWs. It is observed from the Figs. 3 6 that for large value of τ i.e. for τ = 20 the cylindrical and spherical SHWs are similar to one-dimensional 1D) SHWs. Because for a large value of τ the term ν/2τ)ψ is no longer dominant. If the value of τ decreases, the term ν/2τ)ψ becomes dominant. It is observed from Figs. 3 6 that as the value of τ decreases, the amplitude of both the cylindrical and spherical SHWs increases, and the basic features of nonplanar SHWs significantly differ from those of 1D planar ones. It is also seen that the amplitude of the PA SHWs in spherical geometry is larger than those

5 758 Communications in Theoretical Physics Vol. 63 in cylindrical geometry. We have seen that the nonlinear coefficient A decreases exponentially with the increase of superthermal plasma parameters viz. κe, κp, etc.). These are obvious from Fig. 1. The effects of positron and electron superthermality on the phase speed of PA SHWs are displayed in Fig. 2. Fig. 6 Color online) Effects of spherical ν = 2) geometry on PA negative SHWs for µph = 0.1, µe = 0.5, 5 Discussion Fig. 3 Color online) Effects of cylindrical ν = 1) geometry on PA positive SHWs for µph = 0.25, µe = 0.5, Fig. 4 Color online) Effects of cylindrical ν = 1) geometry on PA negative SHWs for µph = 0.1, µe = 0.5, Fig. 5 Color online) Effects of spherical ν = 2) geometry on PA positive SHWs for µph = 0.25, µe = 0.5, The effects of cold positron kinematic viscosity η) on the width ) of the PA SHWs are shown in Fig. 7. The effects of electron and positron superthermality i.e. κe and κp ) on the amplitude of the PA SHWs are shown in Fig. 8. We have investigated the existence of PA SHWS in an unmagnetized, collisionless, non-maxwellian e-p-i plasma consisting of immobile positive ions, cold mobile positrons, and superthermal kappa distributed) hot positrons and electrons. The basic properties of the PA SHWs viz. amplitude, width, polarity, and phase speed) have studied by deriving and analyzing mbe. The effects of cylindrical and spherical geometries and superthermal plasma parameters viz. κp, κe, etc.) on the PA SHWs are numerically examined. It is shown that these parameters viz. κp, κe, σ, µph, µe, and ν) play a significant role in modifying the properties of PA SHWs. The results, which have been obtained from this theoretical investigation, can be pin-pointed as follows: i) The nonlinear coefficient A which measures the inverse of the amplitude) decreases with the increase of the superthermal parameters viz. κp and κe ) as depicted in Fig. 1. ii) The phase speed of PA SHWs increases decreases) with the increase of electron positron) superthermal parameter κe κp ) as shown in Fig. 2. iii) The positive negative) SHWs are found to exist for µph > µc µph < µc ) as illustrated in Figs iv) Figures 3 6 indicate that as the value of τ increases, the amplitude of the cylindrical ν = 1) and spherical ν = 2) PA SHWs decreases. We note that for a large value of τ the nonplanar term ν/2τ )ψ is not dominant, and the nonplanar SHWs are similar to planar PA SHWs. v) The amplitude of the cylindrical ν = 1) PA SHWs is smaller than that of spherical ν = 2) PA SHWs, but is larger than planar PA SHWs shown in Figs. 3 6). vi) The effects of U0 and η on the width of the PA SHWs are shown in Fig. 7. It is identified that the width of PA SHWs decreases increases) with the increase of U0 η). vii) The effects of κe and κp are found to significantly modify the basic properties particularly amplitude) of the PA SHWs as explored in Fig. 8).

6 No. 6 Communications in Theoretical Physics Fig. 7 Color online) Variation of width of PA shock structures with U0 and η. Here σ = 2, µph = 0.25, µe = 0.5, κp = 3, and κe = Fig. 8 Color online) Variation of amplitude of the positive PA SHWs with κe and κp. Here µph = 0.3, µe = 0.6, and σ = 2. We note that our present theoretical investigation is valid only for small but finite amplitude shock structures, but not for arbitrary amplitude shock structures. To examine the properties of arbitrary amplitude PA SHWs and their instability are also problems of great importance for many space plasma situations, but beyond the scope of the present work. We finally hope that the findings of our current theoretical investigation based on nonplanar geometry should be useful in understanding the salient features of nonlinear electrostatic disturbances in space plasma environments viz. Saturn s magnetosphere,[57] solar wind,[58] ionosphere,[59] lower part of magnetosphere, auroral acceleration regions,[60] supernovas, pulsar environments, cluster explosions, active galactic nuclei, etc.) as well as laboratory plasmas.[61 64] References [1] S. Weinberg, Gravitation and Cosmology, Wiley, New York 1972). [2] M.J. Rees, In the Very Early Universe, Cambridge University Press, Cambridge 1983). [3] V.I. Berezhiani, D.D. Tskhakaya, and P.K. Shukla, Phys. Rev. A ) [4] N. Roy, S. Tasnim, and A.A. Mamun, Phys. Plasmas ) [5] K. Roy, A.P. Misra, and P. Chatterjee, Phys. Plasmas ) [6] W. Masood, A.M. Mirza, and M. Hanif, Phys. Plasmas a) [7] F.C. Michel, Rev. Mod. Phys ) 1. [8] R. Saeed and A. Shah, Phys. Plasmas ) [9] H.R. Miller and P.J. Witta, Active Galactic Nuclei, Springer, Berlin 1987). [10] S.K. El-Labany, W.M. Moslem, and E.I. El-Awady, AIP Conf. Proc ) 111. [11] H.R. Pakzad, J. Fusion Energ a) 443. [12] H.R. Pakzad, Can. J. Phys b) 961. [13] I.V. Moskalenko and A.W. Strong, Astrophys. J ) 694. [14] C.M. Surko, et al., Rev. Sci. Instrum ) [15] R.G. Greaves and C.M. Surko, Phys. Rev. Lett ) [16] W. Masood, N. Jehan, A.M. Mirza, and P.H. Sakanaka, Phys. Lett. A b) [17] W. Masood, N. Imtiaz, and M. Siddiq, Phys. Scr a) [18] W. Masood and H. Rizvi, Phys. Plasmas b) [19] A. Shah, R. Saeed, and M. Noaman-ul-Haq, Phys. Plasmas ) [20] P. Chatterjee, D.K. Ghosh, and B. Sahu, Astrophys. Space Sci ) 261. [21] Ata-ur-Rahman, S. Ali, A.M. Mirza, and A. Qamar, Phys. Plasmas ) [22] V.M. Vasyliunas, J. Geophys. Res ) [23] E.I. El-Awady, S.A. El-Tantawy, W.M. Moslem, and P.K. Shukla, Phys. Lett. A ) [24] T.K. Baluku, M.A. Hellberg, I. Kourakis, and N.S. Saini, Phys. Plasmas ) [25] S. Sultana, I. Kourish, N.S. Saini, and M.A. Helberg, Phys. Plasmas ) [26] S. Devanandhan, S.V. Singh, and G.S. Lakhina, Phys. Scr ) [27] S.A. El-Tantawy, N.A. El-Bedwehy, and W.M. Moslem, Phys. Plasmas ) [28] M.S. Alam, M.M. Masud, and A.A. Mamun, Astrophys. Space Sci ) 245. [29] M.S. Alam, M.M. Masud, and A.A. Mamun, Chin. Phys. B a) [30] M.S. Alam, M.M. Masud, and A.A. Mamun, Plasma Phys. Rep b) [31] P.K. Shukla, A.A. Mamun, and L. Stenflo, Phys. Scr ) 295. [32] H.R. Pakzad, Astrophys. Space Sci ) 169. [33] N.A. El-Bedwehy and W.M. Moslem, Astrophys. Space Sci ) 435.

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