arXiv:1707.01757v1 [physics.plasm-ph] 5 Jul 2017 fPyis aagraa nvriy aa,Daa Bangl Dhaka, Savar, University, Jahangirnagar Physics, of afdlItrainlUiest,Damni Dhaka-120 Dhanmondi, University, International Daffodil n pe atninshr 2] yepoignonthermal employing around [26], By [24], magetosphere [27]. of region ionosphere part Martin energy lower upper high- [25], and the shock bow in Earth’s tail charac the long are follow by not distribution) does terized plasmas Maxwell-Boltzmann of prominent least kind such at the of means component the that of equilibrium, one thermodynamic whic positrons, from non- and excess absent electrons of as such (present have particles plasma Maxwellian them nonthermal envi astrophysical a of in generally But few ments species. A plasma [19]–[23] considered plasmas. distribution their ion- e-p-i Maxwellian the isothermal in considered studied [18] have wav [17], electron-acoustic researchers (EAWs) and [14]–[16] many (IAWs) waves plasmas, acoustic of kind [13]. o [12], early of atmospheres center [6], solar [10], surroundi and [5], [9], disc [11], magnetosphere accretion galaxy dwarfs pulsar the [8], white of holes black regions [4], inner p stars [7], etc.), supernovas, universe neutron as [1]–[3], of (such explosions universe regions cluster our environments, of regions pulsar various e-p-i in of existence mas the m disclosed because observations plasmas painstaking (e-p-i) electron-positron-ion of beauty ohhteeto n oirnseis a entheoretica been has Schr nonlinear species) the pressure force) positron deriving (nonthermal (restoring and by species investigated electron inertia positron hot the cold both which the in by provided waves, po (PA) the a of propagation well acoustic nonlinear as The considered. dist positrons is nonthermal function cold Cairns’ following inertial positrons and ions, electrons positive immobile of t,Evlp solitons. Envelope ity, ueo lsaseis,wihcrepn odfeetreal different to correspond situations. which tempe plasma species), modified and space plasma densities significantly of number are (viz. ture gro solitions parameters with envelope plasma regime PA related unstable the and of a regime rate) solitons stable that envelope observed associa (viz. (bright) also properties is solitons dark It the envelope (unstable). stable that bright modulationally t and and waves, supports PA dark consideration with both under equa system of NLS this plasma existence of space analysis numerical the the that from found is It tion. ouainlIsaiiyadGnrto fEnvelope of Generation and Instability Modulational orsodn uhrsemi:[email protected] e-mail: De author’s Educational Corresponding General of Department Depart the with the is with Hossen are R. Mamun M. A. A. and Mannan, A. Chowdhury, A. N. oudrtn h hsc fcletv rcse nsuch in processes collective of physics the understand To natural the by mesmerized are physicists the Nowadays, Abstract ne Terms Index Afu opnn pc lsasse (consisting system plasma space component four —A oiosi orCmoetSaePlasmas Space Component Four in Solitons Psto-cutcwvs ouainlinstabil- Modulational waves, —Positron-acoustic .I I. NTRODUCTION .A hwhr,A ann .R osn n .A Mamun A. A. and Hossen, R. M. Mannan, A. Chowdhury, A. N. dne NS equa- (NLS) odinger ¨ ,Bangladesh. 7, adesh. om velopment, h basic the ribution sitron- hot s plas- for , ron- ment istic olar tion wth is h any ted ra- ng by lly he es ur re of is - oiosaedvtdi e.V ocuini rsre in preserved is Conclusion Envelop V. IV. Sec. Sec. in in sec.VI. presented devoted is are analysis Sec solitons in stability space and The time III. in evolution whic amplitude equation By slow NLS II. the a Sec. governs derive in we presented technique, are perturbation model using plasma our of equations ing tail”. “non-Maxwellian a distribute in nonthermal positrons of and excess electrons having hot plasma NLS of the a MI deriving in the by study equation solitons envelope to of attempt formation we and work, PAWs cold posit present and our inertial in electrons ions, Therefore, hot immobile distributed knowledge, about (PAWs nonthermal with our waves positrons, out plasmas positron-acoustic of worked unmagnetized of best in been properties the have nonlinear to investigations the Up theoretical positro of temperatures. no concentration ion of parameter, t and amplitude nonthermal on the depends of significantly that effects solitons observed envelope bright and and system dark plasma e-p-i for odtosi - lsa.Zhang certain plasmas. under e-i solitons envelope perturbation in of stabili conditions multiscale formation the and examined a EAWs to the derived of of is and equation use NLS Sultana making a plasmas. technique, by e-p-i [17], q-nonextensive in Kaurakis IAWs of (MI) meth Eslami perturbation reductive (RPM), compo- the four employing By unmagnetized plasmas. an nent in deri equation by double-layers Vries as Korteweg-de well as positr structures (quantum waves) solitary acoustic on effects quantum of influence the 3]aaye h fet finknmtc icst nthe on viscosity Sahu kinematics plasmas. Messekher ion Waves. Shock e-p-i of PA of effects in properties the solitons ef analyzed the on and [32] electrons solitons waves. that of nonthermal solitary studied formation of the [31] sound criteria, Pakzad ion certain method, under of energetic pseudo-potential of nature using population By the a of changes presence electrons the Li how plasmas. understand, e-p-i to of structure nonlinear Cairns and linear study to plasma space of kind such nonthemal model to plasmas system. used space unde be better energetic can for high distribution So this [28]. about [26], Satellites standing observ Viking which and with, Freja obtai acceptable by region, more comparatively particles is nonthermal result populated highly the distri Maxwellian in instead species plasma for distribution h rsn ae sognzda olw.Tebscgovern- basic The follows. as organized is paper present The e frsaceshv enue otemldistribution nonthermal used been have researchers of set A tal. et 2]ue otemldsrbto felectrons of distribution nonthermal used [29] tal. et 3]ivsiae ouainlinstability modulational investigated [34] tal. et 3]suidteMI the studied [35] tal. et 3]examined [33] bution rons. ving fect ned on- od he ke ed ns ty r- h d e 1 ) . 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. 1. (Color online)The Variation of PQ or P/Q with wave number k for different values of β. (a) PQ against k for β, (b) P/Q against k for β. All the figures are generated by using these values, µ1 = 0.2, µ2 = 0.7, σ1 = 3, σ2 = 1.5 and β = 0.5.

where number kMI and the frequency ωMI ). Hence, the nonlinear 3k4 C k2 dispersion relation for the amplitude modulation [36], [37] is C = + 3 , 1 2ω4 ω2 given by 3 k C3k 2 2 2 2 Q 2 C2 = + , ω = P k k 2 Φ . (16) 2ω3 ω MI MI MI − P | o| 4 4   3k 2γ2ω Clearly, if PQ< 0, ωMI is always real for all values of kMI , C3 = 4 2 − 2 2 . 2ω (4k + γ1) 2ω k hence in this region the PAWs are stable in the presence of 3 2 − 3 2vgk + ωk + C6ω small perturbation. On the other hand, when PQ > 0 , the C4 = 2 3 , vg ω MI would set in as ωMI becomes imaginary and the PAWs 2 2 2 k + C6ω are unstable for kMI < kc = 2Q Φo /P , where kc is the C5 = 2 , | | vgω critical value of the wave numberq of modulation and Φo is 2v k3 + ωk2 2γ v2ω3 the amplitude of the carrier waves. The growth rate (Γg) of g − 2 g C6 = 2 3 3 . MI (within this conditions, when PQ> 0 and simultaneously γ1vg ω ω − kMI < kc) is given by Finally, the third harmonic modes (m = 3) and (l = 1) k2 and with the help of equations (10) (14), give a system 2 c (17) − Γg = P kMI 2 1. of equations, which can be reduced to the following NLS | | skMI − equation: Clearly, the maximum value Γg(max) of Γg is obtained at ∂Φ ∂2Φ k = k /√2 and is given by Γ = Q Φ 2. i + P + Q Φ 2Φ=0, (15) MI c g(max) | || 0| ∂τ ∂ξ2 | | The coefficients of dispersion term P and nonlinear term Q are dependent on various physical plasma parameters, such where Φ = φ(1) for simplicity. The dispersion coefficient P 1 as σ1, σ2, µ1, µ2, and β. Thus, these parameters may be is sensitive to change the stability conditions of the PAWs. One 1 ∂v 3 ω2 can recognize the stability conditions of PAWs by depicting P = g = v , 2 ∂k −2 k g P/Q against k for different physical plasma parameters. The and the nonlinear coefficient Q is stability of the profile is depicted in Figs. 1(b) and 2, where it is shown that the variation of the ratio of P/Q versus k for ω3 k2(C + C ) Q = 1 4 +2γ (C + C ) different plasma parameters. When the sign of the ratio P/Q 2k2 − ω2 2 3 6  is negative, the PAWs are modulationally stable, while the sign 2k3(C + C ) +3γ 2 5 . of the ratio P/Q is positive, the PAWs will be modulationally 3 − ω3 unstable against external perturbations. It is clear that both  stable and unstable region for PAWs are obtained from the IV. STABILITY ANALYSIS Figs. 1(s) and 2. When P/Q , the corresponding value → ±∞ The evolution of PAWs are governed by the equation of k(= kc) is called critical or threshold wave number for the (15) essentially depends on the coefficients product PQ. onset of MI. This critical value separates the unstable (P/Q > Let us consider the harmonic modulated amplitude solution 0) from the stable region (P/Q < 0) one. 2 Φ = Φo exp(iQ Φo τ). Following the standard stability The nonthermal parameter plays a significant role to change analysis, one may| perturb| the amplitude by setting Φ = the stability of the PAWs. With the increasing values of β Φˆ + ǫΦˆ exp[i(k ξ ω τ)] + c.c (the perturbation wave (nonthermality), the critical value k is shifted to the higher 0 1,0 MI − MI c 4

a HbL H L 0.6 0.4 0.4

0.2 0.2

0.0 0.0 Q Q

  -0.2 -0.2 P P

Μ1=0.1 -0.4 Σ1=2.0 -0.4 Μ =0.2 Σ =2.5 1 -0.6 1

-0.6 Μ1=0.3 Σ1=3.0 -0.8

0 1 2 3 4 5 0 1 2 3 4 5 k k HcL HdL 0.10 0.10

0.05 0.05

0.00 0.00 Q Q  -0.05  -0.05 P P

Μ2=0.3 Σ2=1.0 -0.10 -0.10 Μ2=0.5 Σ2=1.5 0.15 -0.15 Μ2=0.7 - Σ2=2.0

-0.20 0 1 2 3 4 5 6 0 1 2 3 4 5 6 k k

Fig. 2. (Color online) The Variation of P/Q against k for different values of plasma parameters. (a) For µ1, (b) For σ1, (c) For µ2 and (d) For σ2.

HaL HbL HcL 2.0 0.02 0.02

1.5 0.01 0.01 L L F F g 1.0 H 0.00 H 0.00 G Re Re Μ2=0.4 -0.01 -0.01 0.5 Μ2=0.5

Μ2=0.6 -0.02 -0.02 0.0 0 10 20 30 40 50 60 70 -4 -2 0 2 4 -40 -20 0 20 40

K MI Ξ Ξ

Fig. 3. (Color online) Plot of the MI growth rate (Γg ) against kMI for different values of µ2, along with k = 6, and Φ0 = 0.6. (b) Bright envelope solitons for k = 5, (c) Dark envelope solitons for k = 2.2. Along with ψ0 = 0.0005,U = 0.1, τ = 0, and Ω0 = 0.4. value (see Fig. 1(b)). It is seen that the instability sets increases We have also analyzed the effect of effective temperature with the increasing values of nonthermal parameter β. It is also to the hot positron temperature ratio (via σ1) on the stability found that the absolute value of the ratio P/Q increases with of the wave profiles [see Fig. 2(b)]. It is observed that the the increasing values of β. decreasing values of the hot positron temperature (which leads Figure 2(a) shows the variation of P/Q with k for different the increasing values of σ1) the instability sets increases. On the other hand, with the decreasing values of the hot values of hot to cold positron concentration ratio (via µ1) with fixed values of other physical parameters. The critical value electron temperature (which leads the increasing values of of wave number at which the instability sets increases with σ2) the instability sets decreases [see Fig.2(d)]. So increasing hot positron or hot electron temperature plays simultaneously the increasing values of µ1. Actually, increasing the values of hot to cold positron concentration ratio is responsible for the opposite role to recognize the stability region for the PAWs to decreasing values of the nonlinear coefficient Q. As the reason increase or decrease. of the values of decreasing nonlinear coefficient the instability The variation of P/Q with k for different values of hot sets increases. On the other hand, the absolute value of the electron to cold positron concentration ratio (via µ2) with ratio P/Q increases with the increasing values of µ1. fixed values of other physical parameters is depicted in Fig. 5

2(c). It is seen that the critical wavenumber decreases with Figure 3(c) represents the dark envelope solitons. the increasing values of µ2. This leads that the increasing of electron concentration, the critical value (kc) is shifted to the VI. CONCLUSION lower value. So excess number of electron of the system is We considered an unmagnetized four component e-p-i caused to minimize the stability of the wave profile. Again, the plasma system consisting of immobile positive ions, iner- absolute value of the ratio P/Q decreases with the increasing tial mobile cold positrons, and nonthermally distributed hot values of µ . 2 positrons and hot electrons. The well-known reductive per- The variation of MI growth rate (via Γ ) versus MI wave g turbation method has been used to drive a NLS equation, number (via k ) is depicted in Fig. 3(a). It is observed that MI which is valid for a small but finite amplitude limit. We have the growth rate increases with the increasing values of hot observed the existence of both stable and unstable regions of electron to cold positron concentration ratio (via µ2). This outcome also implies that the greater (lower) the values of PAWs and how the related physical plasma parameters (hot positrons temperature, hot electron temperature, cold electron hot electron (cold positron) concentration, the nonlinearity number density, positron number density and nonthermallity) of the PAWs is stimulated (depressed) which expose via the influence to change the stability conditions of PAWs, MI maximum value of MI growth rate. So µ plays a vital role 2 growth rate, and formation of envelope solitons. Like with to the stability of PAWs profile. the increasing of hot positron (hot electron) concentration, the V. ENVELOPE SOLITONS critical value is shifted to the higher (lower) values of k. It If PQ< 0, the modulated envelope pulse is stable (in this is needed to highlights here that the findings of our present region, dark envelope solitons exist) and when PQ > 0, the investigation should be useful for understanding the striking modulated envelope pulse is unstable against external pertur- features of space environments (like cluster explosions, active bations and leads to generation of bright envelope solitons. galactic nuclei, auroral acceleration regions, lower part of The variation of PQ versus k is depicted in Fig. 1(a) with magnetosphere, ionosphere etc.) and laboratory plasmas. different values of nonthermal parameter β. It is found that with the increasing values of nonthermal parameter β, the ACKNOWLEDGMENT values of critical wave number (when PQ = 0) increases. A N. A. 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