Effect of high relativistic ions on ion acoustic solitons in electron-ion-positron plasmas with nonthermal electrons and thermal positrons

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1 Astrophys Spae Si (20) 333: DOI 0.007/s ORIGINAL ARTICLE Effet of high relativisti ions on ion aousti solitons in eletron-ion-positron plasmas with nonthermal eletrons and thermal positrons Kurosh Javidan Danial Saadatmand Reeived: 28 September 200 / Aepted: 7 February 20 / Published online: 26 February 20 Springer Siene+Business Media B.V. 20 Abstrat Propagation of ion aousti solitary waves are studied in e-p-i plasmas ontaining high relativisti ions, Maxwell Boltzmann distributed positrons and nonthermal eletrons. Redutive perturbation method is used and the Korteweg-de Vries (KdV) equation is derived. The effets of high relativisti ions and nonthermal eletrons on soliton haraters are studied. Keywords High relativisti ions Ion aousti solitons Eletron-ion-positron plasmas Nonthermal eletrons Thermal positrons Introdution Solitary waves have been investigated both theoretially and experimentally in most of the branhes of siene. Loalized solutions of nonlinear equations are also extensively studied as important objets in plasma physis (Shukla 2003; Misra and Bhowmik 2007; Verheest 996; Tian and Gao 2007). The ion aousti soliton (IAS) is one of the most aspets of nonlinear phenomena in modern plasma physis researhes. They arise due to a fine tuning between nonlinearity and dispersion effets in the media. Investigation of nonlinear strutures is arried out usually by adopting some forms of perturbation method. In small amplitude approximation of the equations, one ends up deriving some forms of nonlinear differential equations for one spatial dimension situations K. Javidan ( ) D. Saadatmand Department of Physis, Ferdowsi University of Mashhad, Mashhad, Iran Javidan@um.a.ir D. Saadatmand da_se643@stumail.um.a.ir like Korteweg-de Vries (KdV), modified Korteweg-de Vries (m-kdv), nonlinear Shrodinger equation and et. Suh these equations have well known solutions with extended strutures, like solitary wave or solitoni solutions. A great number of authors have been studied ion-aousti solitons using the redutive perturbation tehnique in different situations of plasma physis (Bharuthram and Shukla 986; Yadav and Sharma 99). In ontrast to the usual plasmas that onsisting of eletrons and positive ions, it has been observed that the nonlinear waves in plasmas whih ontaining additional omponents suh as positrons behave differently (Shukla et al. 986). The behaviour of the eletron-positronion plasmas helps us to find better knowledge about the early universe whih assumes to be a kind of plasma (Rees 983; Misner et al. 973), desribing the ative galati nulei (Miller and Witta 987), pulsar magnetospheres (Mihel 982) and also the solar atmosphere (Goldreih and Julian 969). The positrons an be used to probe partile transport in tokomaks and sine they have suffiient lifetime. In this ase, two-omponent (e-i) plasmas beome a three-omponent (e-i-p) medium (Shukla and Stenflo 993; Tsintsadze et al. 994). During the last deade, e-p-i plasmas have attrated the attention of several authors (Popel et al. 995; Lakhina and Verheest 997; Pakzad 2009). We know that when the ion veloity approahes the veloity of light, relativisti effets may signifiantly modify the behaviour of the solitary waves. Relativisti plasmas our in a variety of situations, suh as, spae plasmas (Grabbe 989), laser plasma interation (Arons 979), plasma sheet boundary layer of earth s magnetosphere (Vette 970). This situation is also used in order to desribing the Van Allen radiation belts (Ikezi 973). Das and Paul (985) have investigated the weakly relativisti effets on ion-aousti wave propagation in one dimension using the KdV equation for old plasmas but without eletron inertia effets. Nejoh (987) has inves-

2 472 Astrophys Spae Si (20) 333: tigated the same situation in the warm plasmas. Kalita et al. (996) have studied the existene of solitons with onsidering the omplete fluid equation for eletrons. El-Labany and Shaaban (995) have onsidered nonlinear ion-aousti waves in weakly relativisti plasmas onsisting of warm ion-fluid with non-isothermal eletrons through the modified equations. Pakzad and Javidan (20) have investigated nonlinear ion-aousti shok waves in weakly relativisti plasmas ontaining nonthermal eletrons. Nejoh and Sanuki (994) also have studied the large amplitude Langmuir and ion-aousti waves in relativisti two fluid plasmas with deriving the pseudo potential. Understanding the behaviour of multi speies plasmas ontaining old or warm ions with Boltzmann distribution has been extensively onsidered for the last few years. However, it has been found that the eletron and ion distributions play the ruial role in haraterizing the physis of the nonlinear wave strutures. They further offer a onsiderable inrease in rihness and variety of the wave motion, whih an be found in plasmas. Moreover, they have signifiant influene on the onditions required for the formation of the waves. Modulation of nonlinear waves in slowly responding ollisional plasmas ontaining nonthermal speies has been investigated in Misra and Roy Chowdhury (2002, 2006). Based on observations of solitary wave strutures with density depletion made by Freja and Viking satellites (Dovner et al. 994), Cairns et al. (995a, 995b) have onsidered a plasma model onsisting nonthermally distributed eletrons and ions and emphasizing the role of this distribution on the haraterization of wave strutures (Mamun 2000; Bandyopadhyay and Das 200; Carins et al. 996). It may be noted that plasmas with different temperatures and masses frequently our in spae environment. Partiularly, two temperature eletrons are very ommon in laboratory and spae plasmas. It is found that nonthermal distributions are ommon features of the auroral zone (Lundin et al. 987; Halletal.99). In this paper, the ion aousti solitary waves in plasmas onsisting of nonthermal eletrons, positrons with Boltzmann distribution and high relativisti ions have been studied. 2 Basi equations and derivation of the KdV equation are governed by the following set of equations n t + (nu) = 0 x (γu) t + u (γu) x 2 φ x 2 = n e n n p + φ x = 0 where n and u are the ion number density and ion fluid veloity respetively. φ and γ are eletrostati potential and relativisti fator respetively. For high relativisti plasmas parameter γ is approximated by its expansion up to term u4 4. ) γ = ( u2 2 u 2 = u4 8 4 (2) In order to onsidering the effets of nonthermal eletrons, the eletron number density n e is modeled as (Bandyopadhyay and Das 200) n e = ( βφ + βφ 2 )e φ (3) 4α where β = +3α and α is a parameter that determines the population of nonthermal (fast) eletrons (Kalita et al. 996). The positrons are assumed to be in thermal equilibrium, with the density of n p = pe σφ (4) In the equation set (), the densities of the plasma speies are normalized by the unperturbed eletron density n e0, the ion veloity is normalized by the ion aousti speed i = T e /m, spae variables are normalized by the eletron Debye length λ D = T e /4πn 0 e 2, time variable is normalized by the eletron plasma period T = m e /4πn e0 e 2 and eletrostati potential is normalized by T e e. Note that p = n p0 n e0 represents the relative positron onentration in e- p-i plasma and σ = T e T p is the ratio of eletron temperature to positron temperature. The redutive perturbation method an be used for investigating the behavior of nonlinear ion aousti waves. The strethed oordinates are defined as follows ξ = ε 2 (x λt), τ = ε 3 2 t (5) where ε is a small parameter whih haraterizes the strength of the nonlinearity and λ is the phase veloity of propagated wave. Dependent variables are expanded as follows () Let us onsider one-dimensional, ollisionless, unmagnetized relativisti plasmas with nonthermal eletrons and thermal positrons. The nonlinear dynamis of the low frequeny ion-aousti solitons in the three omponent plasmas n = ( p) + εn + ε 2 n 2 + u = u 0 + εu + ε 2 u 2 + φ = εφ + ε 2 φ 2 + (6)

3 Astrophys Spae Si (20) 333: By substituting (6)into()using(5) and olleting the terms in the different powers of ε, one an derive the following equations in the lowest order of ε n = ( p)u, u = (λ u 0)( β + pσ)φ, λ u 0 ( p) n = ( β + pσ)φ (λ u 0 ) 2 = p (γ 0 + u2 0 )( β + pσ) For the higher orders of ε, wehave (λ u 0 ) n 2 + n τ + (n u ) + ( p) u 2 = 0 n + u n 2 + n u 2 + n u 2 n 2 τ + u 2 [ γ 0 + u2 0 (λ u 0) ( (λ u 0 ) ( + γ 0 + u2 0 γ 0 + u2 0 2 φ (pσ 2 )φ φ 2 2 ( 3u u3 0 ) u2 ) u τ + φ 2 = 0 )] u u = 0 (β ) pσ 3 φ 3 = 0 where γ = + 3u u and γ = 3u 0 + 5u3 2 0 Finally the KdV equation is derived from (7) and (8)as. φ τ + Aφ φ + B 3 φ 3 = 0 (9) where A = [ (λ u0 ) 3 ( pσ 2 )γ 3 + γ ] 2 2 ( p) (λ u 0 )γ γ 2, B = (λ u 0) 3 γ 2( p) The stationary solution of (9) is given by (7) (8) (0) 2 (ξ Uτ) φ = φ 0 se h () w in whih U is onstant veloity of solitary wave. The ion aousti wave amplitude (φ 0 ) and its width (w)aregivenas φ 0 = 3U A, w= 2 B U (2) 3 Disussion and results Equation (8) shows that the oeffiients A and B are funtions of positron density (p), relative temperature (σ ), relativisti fator (η = u 0 ) as well as nonthermal parameter (β). The positron density affeted on A and B through the density ratio p. Some researhers have studied one-dimensional (Das and Paul 985; Nejoh 987) and two-dimensional (Nejoh and Sanuki 994) ases of ion aousti solitary waves in weakly relativisti plasmas ontaining ions and eletrons and the results of their studies are omparable with the presented results of this paper. The above results are also ongruent with Gill et al. (2007) for weakly relativisti plasma with the Boltzmann distribution of eletron (β = 0). The obtained results orretly redue to Nejoh (987) for eletronion (p = 0) weakly relativisti plasmas with Boltzmann distributed eletrons. The above results also are in agreement with the results of Kaur et al. (2009) for eletron-ion (p = 0) weakly relativisti plasmas with nonthermal eletrons. We use the numerial omputation for doing quantitative analysis on the results. Relativisti effet influenes on the equations through the parameter η = u 0. The relative values of the plasma parameters are used to haraterizing the existene of solitons of different types. It is interesting to ompare presented results with those of Gill et al. (2007). Similar oeffiients A and B whih have been appeared in Lundin et al. (987) (with some simplifiations) are as follows: A = [ (λ u0 ) 3 ( pσ 2 )γ ( p) (λ u 0 )γ B = (λ u 0) 3 γ 2( p) γ 2 ] γ 2, (3) where γ = + 3u2 0 and γ = 3u 0. Therefore differenes between our results and the results of Gill et al. (2007) with 2 β = 0 beome A = A A = 2( + p2 σ 2 ) + p( + σ 2 + 6σ) ( p)( + pσ) 3 γ 3 B = B B = p ( + pσ) 3 ( 3u γ ) ( 3u 4 ) (4) Equations (4) learly show that both parameters A and B find smaller values in omparison with similar parameters of Gill et al. (2007); however differenes are very small. Therefore the soliton height and its width beome larger when soliton veloity approahes the veloity of light.

4 474 Astrophys Spae Si (20) 333: Fig. The parameter B as a funtion of relativisti parameter η with different values of β Fig. 3 The soliton height φ 0 as a funtion of relativisti parameter η with different values of p and β = 0. This figure is omparable with Fig. of Gill et al. (2007) Fig. 2 The parameter A as a funtion of relativisti parameter η with different values of β Fig. 4 The soliton height φ 0 as a funtion of relativisti parameter η with different values of p and β = 0.3 Now we an investigate the behavior of soliton amplitude and its width propagated in this kind of plasma medium. Equation (2) shows that Soliton width (amplitude) has a proportional growth with B (/A). Figure presents B as a funtion of η = u 0 for different values of β. This figure shows that B dereases when η inreases. This means that Soliton width dereases when ion veloity approahes to light veloity. B also inreases with an inreasing β. Thus soliton width inreases when the plasma ontains fast eletrons. Figure 2 shows A as a funtion of η for different values of β. This figure demonstrates that A find positive and negative values. Therefore both positive and negative solitons an be reated in this plasma medium. This figure also shows that A has small hanges respet to η. On the other hand, both positive and negative values of A dereases when η inreases. Thus soliton amplitude dereases with an inreasing η. Figure 3 presents the amplitude of soliton as a funtion of η for different values of p with β = 0. This figure shows that our results in the interval 0 <η<0.2 (for weakly relativisti plasmas) onfirm the results of Gill et al. (2007). Equations (4) desribe differenes between we have extrated the numerial results of Gill et al. (2007) by digitizing figures in this paper. It is observed that the hange rate of φ 0 in the high relativisti limit is more than that of in weakly relativisti situation. In omparison with Fig. of Gill et al. (2007), we haven t plotted the profile of φ 0 for p = 0.000; beause it is very lose to urve p = Figure 4 shows the same funtions of Fig. 3 but with β = 0.3. Comparing the soliton amplitude (φ 0 ) in Figs. 3 and 4 shows that the amplitude of soliton in the existene of nonthermal eletrons is inreased. This result also is in agreement with Fig Conlusion and remarks Propagation of solitary waves in ollisionless, unmagnetized high relativisti plasmas with nonthermal eletrons and ther-

5 Astrophys Spae Si (20) 333: mal positrons has been studied. Maximum amplitude of the solitary wave and its width have been derived as funtions of plasma parameters. It is shown that both positive and negative solitons an be reated in this plasma medium. The soliton amplitude inreases when relativisti parameter η inreases. Therefore the amplitude of soliton in the existene of nonthermal eletrons is inreased. But soliton width dereases with an inreasing η. The results have been ompared with the results of weakly relativisti plasmas. We an find a good agreement between the results of these situations. As Fig. 2 learly shows, it is possible that parameter A beomes zero in some speial set of plasma haraters. In this ondition, soliton maximum amplitude goes to infinity. It is lear that our formulation of the problem fails to desribe the situation. This speial state an be investigated in further studies. Plasmas ontaining of positrons, high relativisti ions and also relativisti eletrons is an interesting medium whih has not studied yet. Investigating the propagation of solitary waves in this medium an help us to find better knowledge about the effets of relativisti partiles in plasmas. The results of suggested study an be ompared with the results of Kalita and Das (2007). Referenes Arons, J.: Spae Si. Rev. 24, 47 (979) Bandyopadhyay, A., Das, K.P.: Phys. Sr. 63, (200) Bharuthram, R., Shukla, P.K.: Phys. Fluids 20, 324 (986) Carins, RA, Mamun, A.A., Bingham, R., Bostrom, R., Dendy, R.O., Nairn, C.M.C., et al.: Geophys. Res. Lett. 22, (995a) Carins, R.A., Bingham, R., Dendy, R.O., Nairn, C.M.C., Shukla, P.K., Mamun, A.A.: J. Phys. IV (Frane) 5C6, (995b) Carins, RA, Mamun, A.A., Bingham, R., Shukla, P.K.: Phys. Sr. T63, (996) Das, G.C., Paul, S.N.: Phys. Fluids 28, 823 (985) Dovner, P.O., Eriksson, AI, Bostrom, R., Holbak, B.: Geophys. Res. Lett. 2, (994) EL-Labany, S.K., Shaaban, S.M.: J. Plasma Phys. 53, 245 (995) Gill, T.S., Singh, A., Kaur, H., Saini, N.S., Bala, P.: Phys. Lett. A 364, 367 (2007) Goldreih, P., Julian, H.W.: Astrophys. J. 57, 869 (969) Grabbe, C.: J. Geophys. Res. 94, 7299 (989) Hall, D.S., Chaloner, C.P., Bryant, D.A., Lepine, D.R., Trikakis, J.: Geophys. Res. 96, (99) Ikezi,H.:Phys.Fluids6, 668 (973) Kalita, B.C., Das, R.: Phys. Plasmas 4, (2007) Kalita,B.C.,Barman,S.N.,Goswami,G.:Phys.Plasmas3, 45 (996) Kaur,H.,Gill,T.S., Saini,N.S.:Chaos Solitons Fratals 42, (2009) Lakhina, G.S., Verheest, F.: Astrophys. Spae Si. 253, 97 (997) Lundin, R., Eliasson, L., Hultquist, B., Stastewiz, K.: Geophys. Res. Lett. 4, (987) Mamun, A.A.: Eur. Phys. J. D, (2000) Mihel, F.C.: Rev. Mod. Phys. 54, (982) Miller, H.R., Witta, P.J.: Ative Galati Nulei, p Springer, Berlin (987) Misner, W., Thorne, K.S., Wheeler, J.I.: Gravitation, p Freeman, San Franiso (973) Misra, A.P., Bhowmik, C.: Phys. Lett. A 369, 90 (2007) Misra, A.P., Roy Chowdhury, A.: Fizika A, (2002) Misra, A.P., Roy Chowdhury, A.: Eur. Phys. J. D 39, (2006) Nejoh, Y.: J. Plasma Phys. 37, 487 (987) Nejoh, Y., Sanuki, H.: Phys. Plasmas, 254 (994) Pakzad, H.R.: Astrophys. Spae Si. 323, 345 (2009) Pakzad, H.R., Javidan, K.: Astrophys. Spae Si. (doi:0.007/s ) (20) Popel, S.I., Vladimirov, S.V., Shukla, P.K.: Phys. Plasmas 2, 76 (995) Rees, M.J.: In: Gibbson, G.W., Hawking, S.W., Siklas, S. (eds.) The Very Early Universe. Cambridge University Press, Cambridge (983) Shukla, P.K.: Phys. Plasmas 0, 69 (2003) Shukla, P.K., Stenflo, L.: Astrophys. Spae Si. 209, 323 (993) Shukla, P.K., Rao, N.N., Yu, M.Y., Tsintsadze, K.L.: Phys. Rep. 35, (986) Tian, B., Gao, Y.T.: Phys. Lett. A 362, 283 (2007) Tsintsadze, N.L., Shukla, P.K., Stenflo, L.: Astrophys. Spae Si. 222, 259 (994) Verheest, F.: Spae Si. Rev. 77, 267 (996) Vette, J.I.: Summary of partile population in the magnetosphere. In: Partile and Fields in the Magnetosphere, p Reidel, Dordreht (970) Yadav, L.L., Sharma, S.R.: Phys. Sr. 43, 06 (99)