Modulational Instability and Generation of Envelope Solitons In
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1 Modulational Instability and Generation of Envelope Solitons in Four Component Space Plasmas N. A. Chowdhury, A. Mannan, M. R. Hossen, and A. A. Mamun Abstract—A four component space plasma system (consisting distribution for plasma species instead Maxwellian distribution of immobile positive ions, inertial cold positrons as well as hot in the highly populated nonthermal particles region, obtained electrons and positrons following Cairns’ nonthermal distribution result is comparatively more acceptable with, which observed function is considered. The nonlinear propagation of the positron- acoustic (PA) waves, in which the inertia (restoring force) is by Freja and Viking Satellites [26], [28]. So for better under- provided by the cold positron species (nonthermal pressure of standing about this high energetic space plasmas nonthemal both hot electron and positron species) has been theoretically distribution can be used to model such kind of space plasma investigated by deriving the nonlinear Schrodinger¨ (NLS) equa- system. tion. It is found from the numerical analysis of this NLS equation that the space plasma system under consideration supports the existence of both dark and bright envelope solitons associated A set of researchers have been used nonthermal distribution with PA waves, and that the dark (bright) envelope solitons are to study linear and nonlinear structure of e-p-i plasmas. Like modulationally stable (unstable). It is also observed that the basic Cairns et al. [29] used nonthermal distribution of electrons properties (viz. stable regime and unstable regime with growth to understand, how the presence of a population of energetic rate) of the PA envelope solitions are signiﬁcantly modiﬁed by electrons changes the nature of ion sound solitary waves. related plasma parameters (viz. number densities and tempera- ture of plasma species), which correspond to different realistic By using pseudo-potential method, Pakzad [31] studied that space plasma situations. under certain criteria, the formation of solitons and the effect of nonthermal electrons on solitons in e-p-i plasmas. Sahu Index Terms—Positron-acoustic waves, Modulational instabil- ity, Envelope solitons. [32] analyzed the effects of ion kinematics viscosity on the properties of PA Shock Waves. Messekher et al. [33] examined the inﬂuence of quantum effects on solitary (quantum positron- I. INTRODUCTION acoustic waves) structures as well as double-layers by deriving Nowadays, the physicists are mesmerized by the natural Korteweg-de Vries equation in an unmagnetized four compo- beauty of electron-positron-ion (e-p-i) plasmas because many nent plasmas. By employing the reductive perturbation method painstaking observations disclosed the existence of e-p-i plas- (RPM), Eslami et al. [34] investigated modulational instability mas in various regions of our universe (such as supernovas, (MI) of IAWs in q-nonextensive e-p-i plasmas. Sultana and pulsar environments, cluster explosions [1]–[3], etc.), polar Kaurakis [17], by making use of a multiscale perturbation regions of neutron stars [4], white dwarfs [5], [6], early technique, a NLS equation is derived to examined the stability universe [7], inner regions of the accretion disc surrounding of the EAWs and formation of envelope solitons under certain black holes [8], pulsar magnetosphere [9], [10], center of our conditions in e-i plasmas. Zhang et al. [35] studied the MI galaxy [11], and solar atmospheres [12], [13]. for e-p-i plasma system and observed that the amplitude of To understand the physics of collective processes in such dark and bright envelope solitons signiﬁcantly depends on the kind of plasmas, many researchers have studied the ion- effects of nonthermal parameter, concentration of positrons acoustic waves (IAWs) [14]–[16] and electron-acoustic waves and ion temperatures. Up to the best of our knowledge, (EAWs) [17], [18] in e-p-i plasmas. A few of them have no theoretical investigations have been worked out about considered isothermal Maxwellian distribution [19]–[23], for the nonlinear properties of positron-acoustic waves (PAWs) arXiv:1707.01757v1 [physics.plasm-ph] 5 Jul 2017 their considered plasma species. But in astrophysical environ- in unmagnetized plasmas with immobile ions, inertial cold ments generally a nonthermal plasma (present of excess non- positrons, nonthermal distributed hot electrons and positrons. Maxwellian particles such as electrons and positrons, which is Therefore, in our present work, we attempt to study the MI of absent from thermodynamic equilibrium, that means at least PAWs and formation of envelope solitons by deriving the NLS one of the component of such kind of plasmas does not follow equation in a plasma having excess of nonthermal distributed the prominent Maxwell-Boltzmann distribution) are charac- hot electrons and positrons in a “non-Maxwellian tail”. terized by long tail in the high- energy region [24], around the Earth’s bow shock [25], lower part of magetosphere [26], The present paper is organized as follows. The basic govern- and upper Martin ionosphere [27]. By employing nonthermal ing equations of our plasma model are presented in Sec. II. By using perturbation technique, we derive a NLS equation which N. A. Chowdhury, A. Mannan, and A. A. Mamun are with the Department governs the slow amplitude evolution in time and space in Sec. of Physics, Jahangirnagar University, Savar, Dhaka, Bangladesh. III. The stability analysis is presented in Sec. IV. Envelope M. R. Hossen is with the Department of General Educational Development, Daffodil International University, Dhanmondi, Dhaka-1207, Bangladesh. solitons are devoted in Sec. V. Conclusion is preserved in Corresponding author’s e-mail: [email protected] sec.VI. 2 II. GOVERNING EQUATIONS where vg is the envelope group velocity to be determined later and ǫ (0 <ǫ< 1) is a small (real) parameter. Then we can We consider an unmagnetized four component plasma write a general expression for the dependent variables as system consisting of immobile positive ions, inertial cold positrons, nonthermally distributed hot electrons and hot ∞ ∞ positrons. At equilibrium, the quasi-neutrality condition can be (m) (m) M(x, t)= M0 + ǫ Ml (ξ, τ) exp(ilΘ), expressed as ncp0 + nhp0 + ni0 = ne0, where ncp0,nhp0,ni0, m=1 −∞ X l=X and n are the unperturbed number densities of cold positron, (m) (m) (m) (m) (0) e0 M = [n ,u , φ ]T , M = [1, 0, 0]T , (7) hot positron, immobile ion, and hot electron respectively. The l pcl pcl l l normalized governing equations of the PAWs in our considered plasma system are given by where Θ= kx ωt, where k and ω are real variables represent- − (m) ing the carrier wave number and frequency, respectively. M ∗ l ∂ncp ∂ satisﬁes the pragmatic condition (m) (m) , where + (ncpucp)=0, (1) Ml = M−l ∂t ∂x the asterisk denotes the complex conjugate. The derivative ∂u ∂u ∂φ cp + u cp = , (2) operators in the above equations are treated as follows: ∂t cp ∂x − ∂x ∂2φ = n µ n + µ n µ . (3) ∂ ∂ ∂ 2 ∂ ∂ ∂ ∂ 2 cp 1 hp 2 e 3 ǫv + ǫ , + ǫ . (8) ∂x − − − ∂t → ∂t − g ∂ξ ∂τ ∂x → ∂x ∂ξ For inertialess hot positrons and hot electrons are given by the following expression, Substituting equations (6) (8) into equations (1), (2), and (5) and collecting the power− terms of ǫ, the ﬁrst order (m = 1) 2 2 nhp = (1+ βσ1φ + βσ1 φ ) exp( σ1φ), equation with (l =1) give − n = (1 βσ φ + βσ2φ2) exp(σ φ). (4) e − 2 2 2 iωn(1) + iku(1) =0, iωu(1) + ikφ(1) =0, Substituting equation (4) into equation (3), and expanding up − 1 1 − 1 1 n(1) k2φ(1) γ φ(1) =0. (9) to third order, we get 1 − 1 − 1 1 ∂2φ = n µ + µ µ + γ φ + γ φ2 The solution for the ﬁrst harmonics read as ∂x2 − cp − 1 2 − 3 1 2 3 +γ3φ + , (5) k2 k · · · · · · · · ·· n(1) = φ(1), u(1) = φ(1). (10) 1 ω2 1 1 ω 1 where We thus obtain the dispersion relation for PAWs γ1 = (1 β)(µ1σ1 + µ2σ2), 2− 2 2 2 γ2 = (µ2σ2 µ1σ1 )/2, 2 2 k − 3 3 ω = . (11) γ =(1+3β)(µ σ + µ σ )/6, 2 3 1 1 2 2 (k + γ1) Teff Teff nhp0 and σ1 = , σ2 = , µ1 = , Thp Te ncp0 The second-order when (m = 2) reduced equations with (l = µ = ne0 , µ = ni0 , T = TeThp . 2 ncp0 3 ncp0 eff µ1Te+µ2Thp 1) are In the above equations, the cold positron number density ncp is normalized by its unperturbed number density ncp0 ; 2 (1) (2) k (2) 2ik(vgk ω) ∂φ1 ucp is the cold positron ﬂuid speed normalized by the PA n1 = 2 φ1 + 3 − , 1/2 ω ω ∂ξ wave speed Ccp = (kB Teff /mp) ; φ is the electrostatic k i(v k ω) ∂φ(1) wave potential normalized by kBTeff /e; where kB being (2) (2) g 1 (12) u1 = φ1 + 2− , the Boltzmann constant, Teff being the effective temper- ω ω ∂ξ ature, mp being the positron rest mass, and e being the magnitude of single electron charge. The time and space with the compatibility condition −1 2 1/2 variables are normalized by ωcp = (mp/4πe ncp0) and λ = (k T /4πe2n )1/2 respectively. ∂ω ω(1 ω2) Dp B eff cp0 v = = − . (13) g ∂k k III. DERIVATION OF THE NLS EQUATION The amplitude of the second-order harmonics are found to be proportional to φ(1) 2 To study the modulation of the PAWs in our considered | 1 | plasma system, we will derive the NLS equation by employing the reductive perturbation method. So we ﬁrst introduce the (2) (1) 2 (2) (1) 2 n2 = C1 φ1 , n0 = C4 φ1 , independent variables are stretched as | | | | u(2) = C φ(1) 2, u(2) = C φ(1) 2, 2 2| 1 | 0 5| 1 | ξ = ǫ(x v t), τ = ǫ2t, (6) φ(2) = C φ(1) 2, φ(2) = C φ(1) 2, (14) − g 2 3| 1 | 0 6| 1 | 3 a HbL H L 0.15 0.10 0.005 0.05 0.000 0.00 Q -0.005 -0.05 PQ Β=0.5 P Β=0.6 -0.10 Β=0.5 -0.010 Β=0.7 Β=0.7 -0.15 0.015 Β=0.8 - -0.20 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 k k Fig.