Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions and Non-Isothermal Electrons

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1 Plasma Science an Technology, Vol.17, No.9, Sep. 015 Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions an Non-Isothermal Electrons Apul N. DEV 1, Manoj K. DEKA, Rajesh SUBEDI 1, an Jnanjyoti SARMA 3 1 Department of Science an Humanities, College of Science an Technology, Phuentsholing, Bhutan Department of Applie Sciences, Institute of Science an Technology, Gauhati University, Guwahati , Assam, Inia 3 Department of Mathematics, R. G. Baruah College, Guwahati-78105, Assam, Inia Abstract In this article an investigation is presente on the properties of ust acoustic DA compressive solitary wave propagation in an aiabatic usty plasma, incluing the effect of nonthermal positive an negative ions an non-isothermal electrons. The reuctive perturbation metho has been employe to erive the lower egree moifie Kaomtsev-Petviashivili mk-p, 3D Schamel-Korteweg-e-Vries equation or moifie Kaomtsev-Petviashivili mk-p equations for ust acoustic solitary waves in a homogeneous, unmagnetize an collisionless plasma whose constituents are non-isothermal electrons, singly charge positive an negative non-thermal ions an massive charge ust particles. The stationary analytical solutions of the lower egree mk-p an mk-p equations are numerically analyze, where the effect of various usty plasma constituents on DA solitary wave propagation is taken into account. It is observe that both the ions in usty plasma play a key role in the formation of DA compressive solitary waves, an also the ion concentration an non-isothermal electrons control the transformation of the compressive potentials of the waves. Keywors: usty plasma, reuctive perturbation metho, non-isothermal ions, compressive solitary wave PACS: 5.5.Vy, 5.7.Lw, 5.35.Fp DOI: / /17/9/01 Some figures may appear in colour only in the online journal 1 Introuction Wave an instabilities in low temperature usty plasmas have been wiely stuie in the last few ecaes, since the presence of extremely massive charge ust particles plays an imperative role in unerstaning the electrostatic isturbances in space plasma environments as well as in laboratory plasma evices [1]. The presence of ust grains in a two component electronion plasma is responsible for the appearance of new types of electrostatic waves, incluing solitary or shock waves, an have been reporte by many researchers in both theoretical an experimental points of view [1 11]. One of these electrostatic waves is the low frequency ust-acoustic DA moe in an unmagnetize usty plasma whose constituents are charge ust flui an Boltzmann istribute electrons an ions. It was Rao et al. [1] who were the first to report theoretically the existence of these DA solitary waves SWs. They showe the formation of a rarefactive type of DASWs solution in usty plasmas, an the preictions of DA solitary waves were conclusively verifie by laboratory experiments [13]. In continuation with this, Mamun et al. [14] investigate the nonlinear DA waves in a two component unmagnetize usty plasma consisting of a negatively charge col ust flui an isothermal electrons. Lin et al. [15] consiere the non-thermal ions in a usty plasma to erive a Korteweg-e Vries KV equation for DA waves, an it was foun that the nonthermal ions have a very important effect on the propagation of DA solitary waves. In another work of Mamun [16] it was reporte that the aiabatic effect of inertialess electron an ion fluis has significantly moifie the basic properties of DA solitary waves. In continuation of these, a number of theoretical investigations [17 0] have been mae on DASWs by assuming a three components unmagnetize usty plasma consisting of a negatively charge col ust flui an inertialess isothermal electron an ion fluis. These works are vali only for a col ust flui with isothermal electrons an ions. It has also been confirme from both theoretical an experimental observations that the presence of negative ions in usty plasma plays an important role in many aspects, incluing the charging of the ust particles. In the experimental work of Ahikary et al. [10] the formation of rarefactive DIAWs uner influence of negative ions in a usty plasma was presente together with their characteristic properties. In another stuy of Roy et al. [1] on the role of negative ions with ust charge 71

2 Plasma Science an Technology, Vol.17, No.9, Sep. 015 fluctuation, by estimating the ispersion relation for a DA wave it was shown that the low temperature negative ions can reuce both frequency an amping of the ust acoustic waves. Some numerical investigations on linear an nonlinear DAW showe a significant amount of ions being trappe in the wave potential, which implies that there is a eparture from Boltzmann ion istribution an one woul encounter vortex-like ion istribution in phase space [ 5]. On the other han, the nonlinear behavior of electrostatic waves in a plasma with this non-isothermal state [6 9] has receive consierable attention an has been stuie by a number of authors in the last few years in the context of unmagnetize an magnetize plasmas [30,31]. Mamun et al. [4] investigate the effects of vortex-like an non-thermal ion istributions within the small amplitue regime by using the moifie K-V equation an they conclue the possibility of the coexistence of large amplitue rarefactive as well as compressive ust-acoustic solitary waves, whereas these structures appear inepenently when the wave amplitues become infinitely small. Dorranian an Sabetkar [3] reporte that by increasing nonthermal ion population, the amplitue of the solitary wave ecreases, while the with of solitary waves is increase. Recently Dev et al. [33] also stuie the effect of non-thermal electrons an vortex-like electron istributions in compressive, rarefactive solitary waves an spiky solitary waves in a clo usty plasma by eriving the K-P equation an mk-p equations. Very recently Ahikary et al. erive [34] the mkv equation in a warm usty plasma containing nonisothermal electrons an non-thermal positive an negative ions. So far as it is concerne, the solitary wave in an unmagnetize warm usty plasma containing nonisothermal electrons an non-thermal positive an negative ions has not yet been stuie in etaile 3D form of mk-v an Schamel-K-V equation. In this paper, a etaile investigation of the propagation characteristics of a ust acoustic wave in an unmagnetize warm usty plasma containing non-isothermal electrons an non-thermal positive an negative ions is reporte with erivation of the lower egree moifie K-P, 3D form of Schamel-K-V or mk-p equation. Here the effects of non-thermal ions as well as the non-isothermal electrons are taken into account, an their thermal effects are consiere in an analytical treatment for the small amplitue compressive solitary wave limit. Basic an normalize equations In the present plasma moel we consier an electrostatic ust acoustic solitary wave with extremely low phase velocity which is followe by the negatively charge massive ust particles. Here the pressure provies the restoring force an the inertia comes from the ust mass. The ynamics of the ust particles in a one imensional ust acoustic wave in such a usty plasma system can be escribe by the following basic equations n t. n v 0, 1 t v v 1 p q ϕ, m n m t v p γp v 0, 3 ϕ 4πe n e n n n p Z n. 4 The non-thermal number ensity of positive ion n p, negative ion n n can be escribe by the following relations, n p n p0 1 αφ αφ exp z p φ, 5 n n n n0 {1 ασ p φ α σ p φ } exp z n σ p φ, 6 with α 4γ 1 /1 3γ 1 an σ p T p /T n, where n is the number ensity of the negatively charge stationary ust particles in plasma, Z is the number of electrons resiing on the ust surface at equilibrium, p is the pressure of the ust flui, e is the electron charge, m p m n is positive negative ion mass, z p z n is positive negative ion charge state, v is ust flui velocity, T e T p is electron positive ion temperature, κ B is Boltzmann constant, ϕ is electrostatic potential, γ 1 is the population of non-thermal ions in the plasma. The aiabatic inex γ 5/3 [ D/D, D is the number of egrees of freeom] is ue to the three imensional geometry of the system. The non-isothermal electrons n e can be escribe by the following relations n e n e0 {1 βφ b βφ 3/ 1 } βφ..., 7 with β T p /T e, b 41 γ /3 π > 0, where the parameter γ is efine as γ T ef /T et, in which T ef an T et are temperatures of free electrons an trappe electrons in the plasma, respectively. The parameter γ etermines the nature of the istribution function, giving plateau if γ 0 an a ip if γ < 0, an a hump shape forme if γ > 0. However, γ 1 correspons to the Maxwellian istribution of the electrons. In the present plasma system the range of γ will be consiere as 0 < γ < 1 for the non-isothermal electrons. Now, consiering N ust number ensity normalize by its equilibrium value n 0, V is the ustflui velocity normalize by c s Z κ B T p /m 1/, φ is the DA wave potential normalize by κ B T p ϕ/e, the time variable T is normalize by ω 1 p / m 4πn0 Z e 1/, the space variable X is normalize by λ 1 D 4πn 0 Z e / 1/, κ B T p pressure p is normalize by P n 0 κ B T an from Eqs. 1-4 we get T V N 0, 8 7

3 Apul N. DEV et al.: Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-thermal Ions T V V σ P φ, 9 From the next higher-orer of ε, i.e., ε 7/4, N T V P 5 N 1 3 P V N V V 1 1 V V 0, , τ η ζ φ p 1 φ p φ 3/ p 3 φ N 1, 11 V 1 V V 0 φ P σ 0, 3 with the overall charge neutrality conition τ n e0 n p0 Z n 0 n n0. 1 P 1 P V 0 5 V 5 V 1 1 V 0, τ 3 3 η ζ An µ p n p0 /Z n 0, σ T { /Z T p, µ n n n0 /Z n 0, n e0 /Z n 0 µ e, p 1 µ e β µ n z n } ασ p µ p α z p, p bµ e β 3/, p 3 βµ e / µ n z n/ ασ p z n ασ p µ p z p/ αz p α are consiere. 3 Derivation of lower egree moifie K-P equation Now, to erive the lower egree mk-p equation for the propagation of small but finite amplitue DASW, we use the stanar reuctive perturbation technique in which the inepenent variables ξ an τ are stretche as ξ ε 1/4 X V 0 T, η ε 1/ Y, ζ ε 1/ Z an τ ε 3/4 T, V 0 is the phase spee normalize by c s of the wave along the x-irection an ε is a small nonzero constant measuring the weakness of the ispersion. The epenent variables N, V, P an φ can be expane in power series of ε as N 1 εn 1 ε 3/ N..., 13 V x εv 1 x ε3/ V x..., 14 P 1 εp 1 ε 3/ P..., 15 φ εφ 1 ε 3/ φ ε φ 3..., 16 V y,z ε 5/4 V 1 y,z ε7/4 V y,z Substituting the stretche co-orinates an the expressions for N, V, P an φ into the normalize basic equations Eqs an equating the coefficients of the lowest egree of ε, i. e., ε 5/4 we get, V 1 3V 0φ 1 5σ 3V 18, 0 V 1 η V 1 ζ 3 φ 1 φ 3 p 1 3V 0 5σ 3V0 φ 1 φ 1 η ζ 4, 5 3 p φ 1 1/ N. 6 Substituting the values from Eqs. 18 to 1 into Eqs. to 6, followe by a straight forwar elimination of N, V, P, φ, finally we obtain the following lower egree mk-p equation φ 1 A 1 φ 1 1/ τ φ1 B 3 φ 1 3 φ 1 C η φ 1 ζ 0, 7 where, the nonlinear coefficient A, the ispersion coefficient B an the transverse coefficient C are given by A 1 p p 1 3V 0 5σ 3V, B 0 5σ, C V 0 4V 0 6p 1 V 0. Eq. 7 represents the lower egree mk-p equation which escribes the nonlinear propagation of the DA waves in an electronegative usty plasma with non-thermal ions an non-isothermal electrons. To fin its solution we use the transformation χ lξ mη nζ Uτ an φ 1 ξ, η, ζ, τ ψ χ, then the above equation becomes Bc l 4 ψ χ [ C m n Ul ] ψ A 1l ψ To erive the require solution of the lower egree moifie K-P Eq. 7 we use the well-known tanh-metho an the transformation z tanh χ an ψ χ W z, then Eq. 8 becomes N 1 3φ 1 5σ 3V 0 5φ 1, 19 P 1 5σ 3V0 0. Together with ispersion relation, V 0 [ 5 3 σ 1 ] 1/. {µ p µ n 1βµ n z n ασ µ p α z p } 1 A 1 3 W 3/ l Bc l 4 1 x W x Bc l 4 x 1 x W x { C m n } Ul W 0. 9 For fining the series solution of Eq. 9, substituting W z a r z ρr an using leaing orer analysis r0 of finite terms gives r 4 an ρ 0, an then W z becomes W z a 0 a 1 z a z a 3 z 3 a 4 z 4 { a0 1 z }. Now by referring to the value of W z, 73

4 Plasma Science an Technology, Vol.17, No.9, Sep. 015 the require stationary solution of the lower egree mk- P equation is φ 1 φ m1 sech 4 χ ω 1, 30 where φ m1 5 [ {U l C m n }/ A 1 l ] an [ 64 ω 1 4 Bl 3/{ U l C m n }] 1 are the amplitue an with of the solitary waves, respectively, an l, m, n represent the irection cosines of the angle mae by the propagation irection with the x-axis, y-axis an z-axis an U is the velocity. The non-isothermal effect enhance the amplitue of the lower egree mk-p equation solution but not its with. Since φ m1 is always positive, therefore the solution of the lower egree mk-p equation amits compressive solitions only. 4 Derivation of moifie K-P 3D form of Schamel-K-DV equation For some particular plasma parameters, particularly for Maxwellian electrons if A 1 0 i.e. b 0, then the amplitue of the lower egree mk-p equation φ m1 an the lower egree mk-p equation oes not offer a soliton solution. The lower egree mk-p equation is, therefore, inaequate, an we have to fin another equation in a ifferent egree to stuy the nonlinear properties of the DA waves. So for further amplification we familiarize with b ε 1/ b 1 i.e. b 1 0 Schamel [6 9]. The non-isothermal electrons n e can be escribe by the following relations n e n e0 {1 βφ ε 1 b1 βφ 3/ 1 } βφ.... The normalize Eq. 11 becomes 31 φ p 1 φ p φ 3/ p 3 φ N 1. 3 We use the stanar reuctive perturbation technique in which the inepenent variables ξ, η, ζ an τ are stretching as ξ ε 1/ X V 0 T, η εy, ζ εz an τ ε 3/ T, V 0 is the phase spee normalize by c s of the wave along the x-irection an ε is a small nonzero constant measuring the weakness of the ispersion. The epenent variables N, V, P an φ can be expane in the power series of ε as N 1 εn 1 ε N..., 33 V x εv 1 x ε V x..., 34 P 1 εp 1 ε P..., 35 φ εφ 1 ε φ ε 3 φ 3..., 36 V y,z ε 3/ V 1 y,z ε5/ V y,z Using the above stretching coorinate together with the variable expansion of Eqs in the normalize basic Eqs. 8-10, 3 an collecting the coefficient of the terms of the lowest egree of ε, i.e., ε 3/, we obtain the same result as Eqs However, for the higher egree of ε, i.e., ε 5/, N 1 τ V 1 τ P 1 τ N V 0 V 1 η V V 0 φ P V V 1 η φ 1 p 1 φ p V 1 1 η V ζ V V 1 ζ V 0 N 1 V 1 N 1 0, 38 V 1 V 1 V 1 N 1 φ 1 P σ, 39 V 1 P 1 V 1 ζ 5 3 V 5 3 P 1 V 1 0, 40 φ 1 3 p 3 φ 1 N, 41 3V 0 5σ 3V0 φ 1 η φ 1 ζ. 4 Substituting the values from Eqs. 18 to 1 into Eqs. 38 to 41, followe by a straightforwar elimination of N, V, P, φ, finally we obtain the following new form mk-p equation, φ 1 τ A 1 φ 1 1 φ 1 A φ 1φ1 B 3 φ 1 3 φ 1 C η φ 1 ζ 0, 43 where the ispersion coefficient B an transverse coefficient C are the same as before an nonlinear coefficients A 1, A are erive as A 1 p p 1 3V 0 5σ 4V 0, A 1 V 0 { 7V 0 5σ 6 5σ 3V 0 5σ 3V0 p3 }, 3p 1 with p b 1 µ e β 3/. Eq. 4 represents the mk-p equation which escribes the nonlinear propagation of DA waves in an electronegative usty plasma with non-thermal ions an non-isothermal electrons. The require stationary solution of the mk-p equation is [ φ 1 4A { 1 15Cl Ul C 16A 1 5Cl Ul C 74

5 Apul N. DEV et al.: Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-thermal Ions A } 1 cosh χ, ] 44 ω 3Cl Ul C where ω 1. B l 3 C l U l C The above solution of the mk-p equation epens on the nonlinear coefficient A 1 an A where the nonlinear coefficients A 1 an A can be positive, zero or negative epening on the ifference of the plasma parameter uner ifferent conitions. For negative values of A the above expression gives a complex amplitue, which will not allow the formation of any solitary wave. Therefore the solution of mk-p Eq. 4 gives us compressive solitary waves only when A is positive. Fig. 5 shows the variation of solitary wave profile φ 1 for the lower egree mk-p equation with the variation of non-thermal ion concentration γ 1 an spatial variable χ. Fig. 5 clearly epicts that, with increasing values of the population of non-thermal ion concentration γ 1, the amplitue of compressive solitary waves from the lower egree mk-p equation ecreases. 5 Results an iscussion We analytically examine the epenence of the lower egree mk-p, K-P an higher egree mk-p to explain the propagation of an electrostatic solitary wave for various usty plasma parameters. Here the ust particles in the plasma are consiere as uniform in size an negatively charge, while the backgroun plasma inclues singly charge positive an negative ions. Fig. 1 shows the effect of the variation of positive ion ensity ratio µ p an negative ion ensity ratio µ n on the phase velocity V 0. Here, the other plasma parameters are consiere as m p kg, n e m 3, n p m 3, n n , n m 3, Z e, T e 1.5 ev, T i 0.1 ev [9,10]. Fig. 1 clearly epicts that the phase velocity ecreases with the enhancement of positive ion ensity ratio µ p an negative ion ensity ratio µ n. In Fig., we epict the effect of the variation of ust an ion temperature ratio σ an ions an electron temperature ratio β on the phase velocity λ. The phase velocity ecreases with the ust temperature ratio. Here we consier ust temperature as a constant. That is why the phase velocity ecreases with the temperature of the ions. Similarly it is clear from the figure that the phase velocity increases as the ions an electron temperature ratio β ecreases. Fig. 3 clearly epicts that the amplitue of lower egree mk-p φ m1 increases with the enhancement of positive ions ensity ratio µ p an negative ion ensity ratio µ n It can be conclue from the figure that the lower egree mk-p equation gives the compressive solitary waves whose amplitue increases with the values of positive an negative ions ensity. Fig. 4 clearly epicts that the compressive solitary waves profile of the lower egree mk-p equation ecreases with the enhancement of positive ions temperature ratio β. Here the positive ions temperature ratio increases since we have kept electron temperature as a constant in the ratio i.e. the ion temperature increases as it reflects more ions. We conclue that by increasing the number of ions, uner the same plasmas parameter, the amplitue of the lower egree mk-p equation solution ecreases. Fig.1 The effect of the variation of positive ion ensity ratio µ p an negative ion ensity ratio µ n on the phase velocity V 0 Fig. The effect of the variation of ust an ion temperature ratio σ an ions an electron temperature ratio β on the phase velocity V 0 Fig.3 The effect of the variation of positive ion ensity ratio µ p an negative ion ensity ratio µ n on the amplitue of lower egree mk-p, φ m1 75

6 Plasma Science an Technology, Vol.17, No.9, Sep. 015 in the ratio of the non-isothermal electron concentration γ. i.e. by increasing the trappe non-isothermal electrons temperature uner the same plasmas parameter the solitary waves profile of the lower egree mk-p equation increases. 6 Conclusion Fig.4 The variation of compressive solitary wave profile φ 1 with the ion temperature ratio β for the lower egree moifie K-P equation In the present work we have investigate the properties of ust acoustic DA solitary waves in a warm aiabatic usty plasma in the presence of positive an negative non-thermal ions an non-isothermal electrons. The reuctive perturbation metho is employe to erive the lower egree mk-p equation an the mk-p equation for ust acoustic solitary waves, an the stationary analytical solutions of the lower egree mk-p an mk-p equations are numerically analyze, an the effect of various usty plasma constituents on DA solitary wave propagation is taken into account. It is conclue that, both positive, negative ions ensity ratio, ions temperature ratio, ion concentration γ 1 an electron concentration γ all play an important role in the formation of DA solitary waves. Also we observe that, in the presence of trappe electrons, the compressive solitary wave amplitues ecrease with the enhancement of the number of ions an the compressive solitary wave amplitues increase with the enhancement of the trappe electrons temperature. Fig.5 The variation of compressive solitary wave profile φ 1 with the non-thermal ion concentration γ 1 for the lower egree moifie K-P equation Fig.6 The variation of compressive solitary waves profile φ 1 with the trappe electron concentration γ for the lower egree moifie K-P equation Fig. 6 shows the variation of solitary wave profile φ 1 of the lower egree mk-p equation with the variation of non-isothermal electron concentration γ. This figure reveals that the amplitue of the compressive solitary wave increases with the enhancement of the nonisothermal electron concentration γ. Here the trappe non-isothermal electrons temperature increases since we have kept the free electron temperature as a constant References 1 Goertz C K. 1989, Rev. Geophys., 7: 71 Mamun A A, Shukla P K an Cairns R A. 1996, Phys. Plasmas, 3: 70 3 Misra A P, Chowhury K R an Chowhury A R. 007, Phys. Plasmas, 14: Ali S, Shukla P K. 006, Phys. Plasmas, 13: Saini N S an Kourakis I. 008, Phys. Plasmas, 15: El-Labany S K, El-Taibany W F, Mamun A A, et al. 004, Phys. Plasmas, 11: 96 7 Zhang L P an Xue J K. 008, Phys. Plasmas, 15: Gill T S, Bains A S, Saini N S, et al. 010, Phys. Lett. A, 374: Barkan A, D Angelo N, an Merlino R L. 1994, Phys. Rev. Lett., 73: Ahikary N C, Deka M K an Bailung H. 009, Phys. Plasmas, 16: Shukla P K an Silin V P. 199, Phys. Scripta, 45: Rao N N, Shukla P K an Yu M Y. 1990, Planet. Space Sci., 38: Barkan A, Merlino R L, an D Angelo D. 1995, Phys. Plasmas, : Mamun A A an Hassann M A. 000, J. Plasma Phys., 63: Lin M M an Duan W S. 007, Chaos, Solitons an Fractals, 33: Mamun A A. 008, Phys. Lett. A, 37:

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