We have studied the formation of envelope soliton in a electron-positron plasma. The deduced nonlinear Schrödinger equation yields information about the modulational instability of the system for the different ranges of the plasma parameters. It is important to note that the modulational stability depends on the positron-electron density ratio. The theory is applicable to solitary waves in space plasmas.Proučavamo stvaranje anvelopnih solitona u elektronsko-pozitronskoj plazmi. Izvedena nelinearna Schrodingerova jednadžba daje podatke o modulacijskoj nestabilnosti sustava za razna područja parametara plazme. Važno je primijetiti kako modulacijska nestabilnost ovisi o omjeru gustoće elektrona i pozitrona. Teorija se može primijeniti na solitonske valove u svemirskoj plazmi

Burman, S.

Chowdhury, A. Roy

English

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ISSN1330–0008CODENFIZAE4ON THE ENVELOPE SOLITONS AND MODULATIONAL INSTABILITY INAN ELECTRON – POSITRON PLASMAS. BURMAN and A. ROY CHOWDHURYHigh Energy Physics Division, Department of Physics, Jadavpur University, Calcutta700 032, IndiaReceived 23 June 1998; revised manuscript received 15 March 1999Accepted 4 October 1999We have studied the formation of envelope soliton in a electron-positron plasma.The deduced nonlinear Schrödinger equation yields information about the modu-lational instability of the system for the different ranges of the plasma parameters.It is important to note that the modulational stability depends on the positron-electron density ratio. The theory is applicable to solitary waves in space plasmas.PACS numbers: 52.35.-g, 52.35.Sb, 02.30.Jr UDC 531.742Keywords: electron-positron plasma, envelope soliton, nonlinear Schrödinger equation,modulational stability, dependence on positron-electron density ratio, solitary waves inspace plasmas1. IntroductionIt has been observed recently that nonlinear waves in electron-positron plasmabehave differently from those in plasma with electrons and ions [1]. The electron-positron plasma occured in the early universe [2], and is present in active galacticnuclei [3] and in pulsar magnetosphere [4]. The different nature of an electron-positron plasma is due to the fact that the constituents have almost the same mass.People have already observed small amplitude solitons in plasma with significantpercentage of positrons [5]. Here we study the formation of the envelope solitons inan electron-positron plasma. This is done with the help of a methodology advocatedearlier by Fried and Ichikawa [6] and Roy Chowdhury et al. [7]. In this approach,one first derives a nonlinear dispersion relation which is then utilised to deducethe nonlinear Schrödinger (NLS) equation. The NLS equation so deduced yieldsthe explicit form of the envelope soliton and also yields information about themodulational stability.FIZIKA A (Zagreb) 8 (1999) 2, 83–90 83burman and roy chowdhury: on the envelope solitons and modulational . . .2. FormulationWe consider a plasma consisting of electrons, positrons and positive ions. Itis also assumed that the usual hydrodynamic description is valid. Let nα (α =i,e, p) stand for the ion, electron and positron density and vα (α =i, e, p) betheir respective velocities. Let φ denote the electrostatic potential, (x, t) the spaceand time coordinates, respectively, normalized by kBTe/e, λD =√ε0kBTe/(n0e2)the Debye length, and ω−1i =√ε0mi/(n0e2) the ion plasma period. The ion andelectron velocities are normalized to the sound velocity Cs =√kBTe/mi, where kBis the Boltzmann constant.The continuity equation and equation of motion for the electrons are∂ne∂t+∂∂x(neve) = 0 , (1)(∂∂t+ ve∂∂x)ve − ∂φ∂x= 0 . (2)We assume that the positrons form a background with density np,np = α exp(−φ/β) . (3)The ions are described by∂ni∂t+∂∂x(nivi) = 0 , (4)(∂∂t+ vi∂∂x)vi +1Q∂φ∂x= 0 . (5)The Poisson’s equation is given by∂2φ∂x2= ne − np − ni . (6)Though the species electron and positron are very similar in nature, yet themain reason for the assumption (3) is to simplify the complexity of the problem.Otherwise, we would have two more equations of motion for the positrons, and thewhole analysis would be very complicated.To study the general problem of slow amplitude variation due to nonlineareffects, we proceed along the line of Fried and Ichikawa [6]. Before proceeding to theactual problem, we discuss in short the basics of the approach under consideration.To start with, we assume the existence of a suitable nonlinear dispersion relationε(k, ω, A) = 0 , (7)84 FIZIKA A (Zagreb) 8 (1999) 2, 83–90burman and roy chowdhury: on the envelope solitons and modulational . . .k being the wave vector, ω the frequency and A the amplitude. Therefore, ε denotesthe functional relation between k, ω and A for the plasma under consideration.When there is no dependence on A, the relation becomes the usual linear dispersionrelation. If initially the wave amplitude of the electrostatic potential φ is φ(x, 0),φ(x, 0) =∫dk φk exp(ikx) + c.c. , (8)with φk being peaked around k = k0. In the usual linear theory, the long-timebehaviour is given asφ(x, t) =∫dk φk exp{i(kx− ωt)} + c.c . (9)In the basic formulation of Fried and Ichikawa [6], it was assumed that one canstill use (9) in some approximation. We consider the situation when the amplitudeand the spread of k values are small. Then, it is meaningful to talk about anexpansion of the dispersion relation both in A2 and k̄ = k − k0. Soε(k, ω, A) = ε(k, ω, 0) +A2∂ε∂A2+ . . . = 0 . (10)For ω, we haveω = Ω(k) +MA2 = ω0 +Γ , (11)andΓ = vg k̄ + v′gk̄22+MA2 , (12)Ω (k) being the solution of the linear dispersion relationε[k,Ω(k), 0] = 0 ,ω0 = Ω(k0), vg =∂Ω∂k∣∣∣∣k=k0, (13)v′g =∂2Ω∂k2∣∣∣∣k=k0, M =∂ω∂A2∣∣∣∣k=k0, ω = ω0, A = 0 .Equations (10) and (11) were used by Fried and Ichikawa [6] to deduce the nonlinearSchrödinger equationi(∂φ∂t+ vg∂φ∂x)+v′g2∂2φ∂x2+ q1φ |φ|2 = 0 , (14)FIZIKA A (Zagreb) 8 (1999) 2, 83–90 85burman and roy chowdhury: on the envelope solitons and modulational . . .q1 = −M = − ∂ω∂A2.To apply the above methodology in the case under consideration, we first of alldeduce the (linear) dispersion relation of the system represented by Eqs. (1) to (6).For this we setne = 1 + n′e, ve = v′e ,np = n0p + n′p, ni = n0i + n′i , (15)vi = v′i φ = φ′ ,where each primed variable denotes perturbed quantity which is proportional toexp{i(kx− ωt)}. Proceeding in the usual manner, we getω2 =1 + (1− α)/Qα/β· k21 + (β/α)k2. (16)So, this is our ε(k, ω, 0). To introduce corrections for small variation of the ampli-tude, we proceed to a frame of reference moving with velocity V and set ξ = x−V t.Whence, upon integration, the equations yieldve = Vne − 1ne− 2V 2ne − 1ne+ V 2(ne − 1)2n2e− 2φ = 0 , (17)vi = Vni − 1 + αni,from which one can solve for ne and nini =1− α√1− 2φ/(QV 2) , ne =1√1 + 2φ/(V 2). (18)The Laplace equation leads to∂2φ∂x2= ne − np − ni = 1√1 + 2φ/V 2− αe−φ/β − 1− α√1− 2φ/(QV 2) . (19)Expanding the right-hand side of Eq. (19) in powers of φ, we get∂2φ∂x2= F1φ+ F2φ2 + F3φ3 , (20)whereF1 =αβ− 1 + (1− α)/QV 2,86 FIZIKA A (Zagreb) 8 (1999) 2, 83–90burman and roy chowdhury: on the envelope solitons and modulational . . .F2 =321− (1− α)/Q2V 4− α2β2, (21)F3 =α6β3− 521 + (1 − α)/Q3V 6.We now try to analyse Eq. (20) by a Fourier decomposition of φ,φ =∞∑−∞φn exp(inkx), φn = φ?−n . (22)Whence, we get a set of equations of the following form−n2k2φn = F1φn + F2∑p+m=nφpφm + F3∑p+m+l=nφpφmφl . (23)Furthermore, it is assumed that each of the Fourier modes φn can be expanded inpowers of ε asφn =∞∑1εpφpn, λ = −k2 =∞∑0εpλp , (24)where ε represents a scale length. We are actually computing the perturbativecorrections to φn and k2 in different orders of ε. Here, for convenience, we have setλ = −k2. Whence, we get in various powers of εφ1±1 = 1, φ(1)n = 0, n /=± 1 ,λ0 = F1 = α/β − {1 + (1− α)/Q}/V 2 ,φ(2)0 = −2F2/F1 , (25)φ(2)±2 = +F2/(3F1) ,λ2 = 2F2(φ22 + φ20)+ 3F3 .From this information one can at once calculate the nonlinear (q1) and disper-sive (v′g/2) coefficients appearing in the nonlinear Schrödinger equation (14). Oneobtains.q1 = −12α/β ·G{1 + (1− α)/Q}(α/β + F1)2 , (26)p1 = +v′g2=12∂2ω0∂k2= −32k√1 +1− αQ√αβ+ k2 . (27)where G = 2F2(φ22 + φ20) + 3F3.FIZIKA A (Zagreb) 8 (1999) 2, 83–90 87burman and roy chowdhury: on the envelope solitons and modulational . . .3. DiscussionIt is now a well-known fact that the modulational instability can be studiedvery simply by looking at the product p1q1 and analysing its variation with re-spect to the plasma parameters. In our case, the most important parameter is αwhich is equal to np0/n0, the ratio between the equillibrium densities of positronsto electrons. In Figs. 1 and 2, we show the variation of p1 and q1 as functions ofα. Two cases may arise, either α > 1 or α < 1. The two situations are exhibitedin Figs. 1a and b, and 2a and b, respectively. The case when α > 1 shows that0 2 4 6 8 10 12 14 160,000,050,100,15(a)p 1α0 2 4 6 8 10 12 14 160500100015002000(b)q 1αFig. 1. Variation of a) p1 and b) q1 (see text) for α > 1.88 FIZIKA A (Zagreb) 8 (1999) 2, 83–90burman and roy chowdhury: on the envelope solitons and modulational . . .0,1 0,2 0,3 0,4 0,5 0,6-100-75-50-250255075100(a)p 1α0,1 0,2 0,3 0,4 0,5 0,60,000,050,100,15 (b)q 1αFig. 2. Variation of a) p1 and b) q1 (see text) for α < 1.both p1 and q1 are of same sign and hence the product p1q1 is always positive. Onthe other hand, for the case α < 1, the situation as depicted in Figs. 2a and b,turns out to be different. Here we observe that for 0.1 < α < 0.2, the function p1is negative and sharply changes to positive as α becomes greater than 0.2. But q1remains positive althrough. So, here we observe a change of stability. Dependingupon the relative concentration of the electrons and positrons, a stable situationmay turn to be an unstable one. Due to the existence of a large amount of electronsand positrons in the space plasma, it is always possible to have a practical situationwhere such conditions can occur.FIZIKA A (Zagreb) 8 (1999) 2, 83–90 89burman and roy chowdhury: on the envelope solitons and modulational . . .References1) F. B. Rizzato, J. Plasma Physics, 40 (1988) 289;2) M. J. Rees, The Very Early Universe, Cambridge University Press, Cambridge (1983);3) H. R. Miller and P. J. Witta, Active Galactic Nuclei, Springer-Verlag, Berlin (1987) p.202;4) F. C. Michel, Rev. Mod. Phys. 54 (1982) 1;5) T. E. Hansen and A. G. Emslie, The Physics of Solar Flares, Cambridge UniversityPress, Cambridge (1988) p. 124;6) D. E. Fried and Y. H. Ichikawa, J. Phys. Soc. Japan 34 (1973) 1073;7) A. Roy Chowdhury, K. Roy Chowdhury and S. N. Paul, Nuovo Cimento D 15 (1993)1043.ANVELOPNI SOLITONI I MODULACIJSKA NESTABILNOST UELEKTRONSKO-POZITRONSKOJ PLAZMIProučavamo stvaranje anvelopnih solitona u elektronsko-pozitronskoj plazmi. Izve-dena nelinearna Schrödingerova jednadžba daje podatke o modulacijskoj nesta-bilnosti sustava za razna područja parametara plazme. Važno je primijetiti kakomodulacijska nestabilnost ovisi o omjeru gustoće elektrona i pozitrona. Teorija semože primijeniti na solitonske valove u svemirskoj plazmi.90 FIZIKA A (Zagreb) 8 (1999) 2, 83–90

Anvelopni solitoni i modulacijska nestabilnost u elektronsko-pozitronskoj plazmi

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