760 S.K. El-Labany et al Vol. a concluding discussion is presented and a coparison with previous results is considered..basic equations and derivation

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1 Vol No 7, July 003 cfl 003 Chin. Phys. Soc /003/(07)/ Chinese Physics and IOP Publishing Ltd Modulational instability of a wealy relativistic ion acoustic wave in a war plasa with nontheral electrons S. K. El-Labany, M. S. Abdel Kri, S. A. El-Warrai, and W. F.El-Taibany y Physics Departent, Faculty of Science-Daietta, Mansoura University, Daietta El-Gedida, Egypt (Received 5 Noveber 00 revised anuscript received March 003) An investigation has been ade of odulational instability of a nonlinear ion acoustic wave in a wealy relativistic war unagnetized nontheral plasa whose constituents are an inertial ion fluid and nontherally distributed electrons. Up to the second order of the perturbation theory, a nonlinear Schrödinger type (NST) equation for the coplex aplitude of the perturbed ion density is obtained. The coefficients of this equation show that the relativistic effect, the finite ion teperature and the nontheral electrons odify the condition of the odulational stability. The association between the sall-wavenuber liit of the NST equation and the oscillatory solution of the Kortewegde Varies equation, obtained by a reductive perturbation theory, is satisfied. Keywords: derivative expansion ethod, war plasa, ion acoustic waves, odulational instability PACC: 535. Introduction The odulation of one-diensional (D) ion acoustic waves (IAWs) in an unagnetized, collisionfree plasa in the finite-wavenuber region has received a great deal of attention. [4] These studies have shown that the aplitude of the perturbed ion density is governed by a nonlinear Schrödinger type (NST) equation. However, these studies were concerned with non-relativistic cold plasas. [57] For an envelope soliton, there has been a great deal of interest in studying the odulational instability of different wave odes in plasa, because of its iportance in stable wave propagation. Experiental observations of the odulational instability of the onochroatic IAW have been reported by Watanabe. [8] On the other hand, ost of the investigations of the propagation of the nonlinear ion acoustic solitons have been developed on the basis of the reductive perturbation theory. In this theory a set of coupled nonlinear partial differential equations describing the syste is reduced to the Kortewegde Varies (KdV) equation or to the NST equation, depending on whether the syste is a wealy dispersive or strongly dispersive ediu. y Corresponding author. E-ail: eltaibany@eail.co Recently,using the reductive perturbation technique, Xue et al [9] study the odulational instability of the odulated IAW in a war plasa. However, this proble had been investigated by El-Labany [4] and El-Labany and El-Hanbaly [0] using the derivative expansion ethod by Kawahara. [] A few years ago, otivated by the observations of solitary structures with density depletions. Cairns and co-worers [4] have considered a plasa consisting of nontheral electrons, with excess of energetic particles and cold ions and have shown that it is possible to obtain both positive (copressive) and negative (rarefactive) solitary waves. In this paper, we study the odulational instability of the odulated wealy relativistic IAW in a collisionless unagnetized war plasa with nontheral electrons, using the derivative expansion technique.it is organized as follows. In section a wealy relativistic NST equation is derived fro the fluid equations describing the syste. In section 3 the stability analysis of this equation is investigated. In section 4 the relationship between wealy relativistic ion odulation odes in the sall wavenuber region and in the finite wavenuber region is investigated. In section 5

2 760 S.K. El-Labany et al Vol. a concluding discussion is presented and a coparison with previous results is considered..basic equations and derivation of the NST equation Consider a siple wealy relativistic plasa odel that includes one war ion species together with nontheral electrons. Also, we assue that the plasa is unagnetized, collisionless and ionizationfree. The one-diensional basic equations can be written in a @t (nu) = + (flu)+ n = 0 () p = 0 ffi = n e n (4) n e = e ffi [ ( )( ffi)ffi] (5) n and n e are the densities of the ions and electrons respectively, u is the flow velocity of the ions, p is the pressure, ffi is the electrostatic potential, ( ff) = deterines the population of nontheral electrons in the present plasa odel, x is the ( + 3ff) space coordinate, t is the tie variable and ff fi is the ratio of ion teperature T i to electron teperature T e. The relativistic factor is approxiated by its expansion up to the second ter because of a wealy relativistic effect, fl ß + u (6) C C is the velocity of light. All physical quantities in Eqs.(5) as well as the space and tie variables have been rendered diensionless in ters of the following quantities: ion theral velocity = ( B T e =) =, ion pressure = n 0 B T i, the Debye length D = ( B T e =4ße n 0 ) =, a characteristic potential ( B T e =e) and a characteristic tie (4ße n 0 =) = (the inverse of the ion plasa frequency! pi =), B is Boltzann's constant and is the ion ass. According to the general ethod of the derivative expansion, [4] we introduce the stretched variables fi i = " i t ο i = " i (x t) (i = :::)ο 0 = x (7a) (7b) the paraeter is the group velocity, to be deterined later and the sallness paraeter " represents the size of the perturbed aplitude. To derive the NST equation describing the propagation of the nonlinear ion acoustic waves fro the basic equations (5), we expand all the quantities asyptotically in the sallness paraeter " about their equilibriu values G(x t) =G o + Here G (L) G Λ(L) G (L) X X L= " G (L) (fi fi :::ο ο :::) (8) = [n (L) u(l) p(l) ffi(l) ] G o = [ u 0 0] and = x!t: satisfies the reality condition G (L) =, the asteris denotes the coplex conjugate. The dependence on the scales fi fi ::: ο ο ::: is chosen in such a way that the secular ters are eliinated. Substituting Eqs.(7) into the basic equations ( 5), we obtain, to first order in " and L = coponents u () = n() p() = 3fl n () and ffi () = n() + (9) fl = + 3 u 0 C =! u 0and the linear dispersion and the group velocity are given respectively by fl = 3fl ff + + (0a) = u 0 + [3fl ff + ]: (0b) fl ( + ) The L = 0 coponents of O (") give n (0) = n (0) e and ffi(0) = n(0) : Turning to O (" )of the reduced equations, we can obtain, for L = 0 (0) = ffi (0) = 0 and ffi (0) = [n(0) ffi Λ() ffi () ] for L = () = 0 u () = n () +

3 No. 7 Modulational instability of a wealy relativistic ion acoustic wave in p () =3fl n () and ffi () = n() () () ( + ) + j ffi (0) n ( + ) = j Z ~ for L = coponents, + 3fl ffb + Z + B and n () = ( + A )n () u () = A n () A = + (3 ) 6 A = fl p () = 3fl (A + B )n () ffi () = (A A ) ( + ) n() + fl fl ffh fl +3fl ff( + 3fl )( + ) H = ( + + 3fl )( + )(4 + ) B = +3fl + fl fl and fl = 3u 0 C : Equating ters of O (" 3 ) in the set of equations (5), we have for the L = 0 coponents, () j n + B u (0) = j C Z () j n (0) n = j C Z ~ p (0) + 3fl ffb + 3fl ffb + Z () =3fl ~ j n j C Z + 3fl ffb Q = fl 8 >< + B + B Z + B 3ff j n() j ( + ) B = ~ ( + [=]) ( + ) fl ~ Z = fl ~ 3fl ff (=), ~ = u 0 and C is a constant of integration, independent of ο. Using the first-and second-order solutions for the L = coponents of the set produced equations of O (" 3 ), the resonant ters leads to the non-secular condition and the NST equation, as () +RC n () +Q j n() j n () = 0: () The coefficients of the linear, dispersive and nonlinear ters are given respectively by R = ( + [=]) 3fl fl ~ ( + ) ff( 3fl B ) + C fl fl fl Z fl ( + ) ( + [=]) + + 3fl ff(3fl ) ~( + ) ~ S = 4 ~ ( + ) = d fl ( 3 + ) 3 d Q = R Q 4 +6A fl +4A fl w 3fl B fl 3 +6fl w 3 3 w 3 w +fl + fl A w fl 4 fl w 3 fl C 3 9 >= >: fl ff + A + + A +A + ( + ) 3 > and C = B fl ffb : (see Appendix A). Equation () governs the evolution of the coplex aplitude of the nonlinear IAW in the finitewavenuber region, propagating in a wealy relativistic war plasa with nontheral electrons. 3. Stability analysis The sign of the dispersion coefficient Q charac-

4 76 S.K. El-Labany et al Vol. terizes the aplitude of the odulated IAWs defined by the NST Eq. (). The stability criterion of this equation exhibits a odulational stability of the aplitude of the wave envelope if SQ < 0, and exhibits a odulational instability if SQ > 0. We now that the forer case gives rise to a dar envelope soliton and the latter gives a bright envelope soliton. However, the expressions of S and Q show that the stable and unstable regions are odified due to the finite ion teperature (ff), relativistic effect (fl fl )and nontheral electron (). Since S is always negative for w > u 0, one has to deterine the value of the critical wavenuber C (ff, fl )at which Q changes its sign. Then, for all values of < C, the wave has a odulational stability, while the odulational instability occurs in the region > C. Applying the dispersion relation and after straightforward anipulation, we can reduce the nonlinear coefficient in the for: X0 w i (0i) i= Q(ff fl ) = fl D( + ) () 4 D = 3( + )( + fl ff )+( 4 =) and w i (i = :::0) is the weight of each ter. Figure shows the dependence of C, in units of D, on the nontheral paraeter. It is shown that C first decreases rapidly, then increases slowly for the higher values of. It is proved that as the ion teperature ff or the relativistic effect (u 0 =C) increase, C decreases but the variation with the relativistic effect is sall. 4. Oscillatory solution of the to KdV equation In the sall-wavenuber region, Eq. () 3 n () 3 n () + a j n () j n () 3 b ρ + 3[ + fl ff +3fl a = ff] p fl ([=]+3fl ff) b =[ p fl (+3fl ff)] : = 0 (3) ff fl fl (see Appendix A). Applying the reductive perturbation theory, the coplex aplitude of the perturbed density in a wealy relativistic war plasa with nontheral electrons in the sall-wavenuber liit is governed by the KdV equation, which is ~n = 0 ~n is the perturbed density, and fi and ο are given by ο = μ (x μ t)fi = μ 3 t: Here μ is the ordering paraeter and the wavenuber = O (μ ). [4] To obtain an oscillatory solution of the KdV Eq. (4), we follow El-Labany [4] and El-Labany and El- Hanbaly, [0] expanding (~n ) as ~n = X X " = L= ~n L (ρ )exp:[il(ο ffifi)] (5) Fig.. The dependence of C (in units of D) on the nontheral paraeter for certain values of relativistic paraeter u0=c = 0 (non-relativistic) and 0:. C first decreases rapidly then increases slowly for higher values of as it approaches the isotheral value ( = ). the stretched variables ρ and are related to ο and fi by ρ = "(ο fifi) = " fi: (6) The paraeters ffi and fi will be deterined later. Substituting Eqs.(5) and (6) into Eq.(4), we obtain the

5 No. 7 Modulational instability of a wealy relativistic ion acoustic wave in following reduced equation of order : iffil~n (L) 3 + ia + X X 0 = L= X X = L 0 = b(l)3 ~n (L) + 3 ~n + b (L L 0 )~n (L) ~n (L0 ) 0 ) 0 ~n(ll0 3 ~n (L) (LL 0 ) 0 = 0: The first-order ters ( = ) with L = ± lead to the linear dispersion relation ffi = b3 : The second-haronic coponents of the second-order ters of the reduced Eq. (7) gives ~n () = a 3b ~n() while the coponents L = ± of this order lead to the copatibility endurance condition for non-trivial solution, i.e. fi = 3 b : The zeroth-haronic coponents of the third-order ters of the th reduced equation are deterined as ~n (0) = a 3 fi (j n() j C ) C is a constant independent of ρ. Finally, the L = ters of the third-order reduced equation give the NST equation (Eq.(3)). In ters of the solution of this equation, the oscillatory solution of Eq.(4) is expressed as ρ r ~n ="~n () +3fl ff x (u 0 + fl 3 p t "t fl [ + 3fl ff] ρ r +3fl ff exp i x u 0 + fl p fl [ + 3fl ff] 5. Conclusions ff To verify our results, we can consider a special case of a wealy relativistic war plasa with isotheral electrons and then ae coparisons with published wors. For = (isotheral electrons), all the t : ff coefficients in this wor agree exactly with those obtained by El-Labany. [4] Also, if we neglect the relativistic effect (put fl = fl = 0), we can acquire the result of Xue Ju-Kui et al Thus, this paper can be considered as the generalization of the wor done by El-Labany [4] with the inclusion of nontheral electrons. We conclude that, on the basis of the derivative expansion ethod, the odulation of nonlinear IAW in a wealy relativistic war plasa with nontheral electrons has been investigated. A NST equation describing the evolution of the coplex aplitude of the perturbed ion density in the finite wavenuber region is obtained. The coefficients of this equation have been shown to be strongly dependent on the ion teperature ff, the ion streaing velocity (fl fl ) and the nontheral paraeter. We can suarize these effects as follows:. The critical wavenuber C decreases as ff increases for fixed values of ion streaing, and also C decreases as the relativistic effect increases but its variation with the latter is slightly slow.. The C first decreases rapidly as increases, then it increases slowly as increases. This eans there exists a critical nontheral population of electrons to sustain the syste in the stable state. Moreover, we have derived the relation between the wealy relativistic ion odulation odes described by the KdV equation in the sall wavenuber region and the NST equation in the finite wavenuber region. The dispersion and the nonlinear coefficients of the KdV equation are exactly the sae as those of the sall wavenuber liit of the NST equation derived in the finite wavenuber region. In addition, the oscillatory solution of the KdV equation satisfies the NST equation. Appendix A To evaluate the coefficients of the NST equation for sall, we first calculate the different ters appearing in these coefficients. Fro Eqs.(0), as! 0, we have thus Z = ß ~ ß +3fl ff fl ( + ) ( f3flff( + )+g ( +3)( + )) ß 3

6 764 S.K. El-Labany et al Vol. the square-bracet notation indicates that the quantity is evaluated at! 0. Also 3 [A] = fl 6 ( ) 4fl 6 3fl ( )( + 4 ) 5 + fl ff f + (+3fl)( + )(4 + )g ß+ 3 + ( + = 3fl ff) 4fl +3fl ff 6 6 3= 3fl ( + 4 )+ fl ff f +(+3fl)(5 + )g and [A] = [B] = fl +3flff( + ) fl ß fl 3= +3fl ff fl +3fl + fl fl ~( + ) [B] = ( + ) ß ~ 3 ( ) fl [C] = + 3fl ffb ß +3fl ff fl + fl fl 3= = +3flff( + 3fl) ß +3fl + ( + 3fl ff) fl ~ +3fl ff fl + B =ρ + 3fl ff =ff +3fl ff + ~ 3 ( ) fl +3fl +3fl ff fl : Thus the coefficients S, R and Q are given by [S] = ß 3 fl ( + ) 3 ρ fl +3fl ff ρ [R] = C fl fl flz fl + ~ ρ 3 ßρ flff(3fl ) + 3fl ff ρ fl +3fl ff 4 ~ ( + ) ff = ffff = 3 b =ff ( + 3fl) fl 3= 3flff + fl ff = ρ ff = [Q] = fl +3fl ff Aρ + fl ff +3flff =ρ 3fl + fl fl + 3fl ff and [Q] = [Q] [R] = [Q]: fl fl (+3fl) fl fl 3= 3flff + =ff References [] Shiizu K and Ichiawa Y H 97 J. Phys. Soc. Japan [] Kautani T and Sugioto N 974 Phys. Fluids 7 67 [3] EL-Labany S K 99 Astrophys. Space Sci. 8 4 [4] EL-Labany S K 995 J. Plasa Phys [5] Ain M R, Morfill G E and Shula P K 998 Phys. Rev. E [6] Maun A A 999 Phys. Scr [7] Xie B S 00 Chin. Phys. 0 4 [8] Watanabe S 977 J. Plasa Phys [9] Xue J K, Duan W-S and Lang H 00 Chin. Phys. 84 [0] EL-Labany S K and EL-Hanbaly A M 995 Il Novo Ciento 7D [] Kawahara T 973 J. Phys. Soc. Japan [] Cairns R A, Maun A A, Bingha R, Dendy R O, Bostro R, Shula P K and Nairn C M C 995 Geophys. Res. Lett. 709 [3] Cairns R A, Bingha R, Dendy R O, Nairn C M C, Shula P K and Maun A A 995 J. Physique VI 5 C6-43 [4] Cairns R A, Maun A A, Bingha R and Shula P K 996 Phys. Scr. T 63 80