Electronegative (3+1)-dimensional modulated excitations in plasmas

Accepted Manuscript Electronegative (3+1)-dimensional modulated excitations in plasmas Conrad B. Tabi, Chérif S. Panguetna, Timoléon C. Kofané PII: S0921-4526(18)30432-0 DOI: 10.1016/j.physb.2018.06.032 Reference: PHYSB 310940 To appear in: Physica B: Physics of Condensed Matter Received Date: 7 April 2018 Revised Date: 3 May 2018 Accepted Date: 25 June 2018 Please cite this article as: C.B. Tabi, Ché.S. Panguetna, Timolé.C. Kofané, Electronegative (3+1)dimensional modulated excitations in plasmas, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.06.032. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Electronegative (3+1)-dimensional Modulated Excitations in Plasmas 1 2 RI PT Conrad B. Tabi1,2∗, Chérif S. Panguetna3†, and Timoléon C. Kofané3‡ Botswana International University of Science and Technology, Private Bag 16 Palapye, Botswana Laboratoire de Biophysique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, B.P. 812 Yaoundé, Cameroun 3 Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, TE D M AN U June 26, 2018 SC B.P. 812 Yaoundé, Cameroun Highlights • Modulated envelope waves are studied in three-dimensional electronegative plasmas. EP • The modulation angle and the plasma parameters are found to affect the stability of the plane wave solutions. AC C • Oblique, parallel and transverse modulations are discussed. • Parallel and transverse modulations are detected in opposite plasma parameter regions. ∗ Corresponding author: [email protected] or [email protected] (C. B. Tabi) [email protected](C. S. Panguetna) ‡ [email protected] (T. C. Kofané) † 1 ACCEPTED MANUSCRIPT Abstract A three-dimensional electronegative plasma model is studied. Modulated ion-acoustic waves are investigated via the activation of modulational instability in the Davey-Stewartson equations, with three space variables. The contributions of the modulation angle (θ) and electronegative plasma parameters are discussed to that effect, including some particular cases such as the RI PT parallel and transverse modulations. The parametric linear stability analysis that is proposed shows that the growth rate of instability displays opposite regions of instability for parallel and transverse modulations when the negative ion concentration ratio (α) and the electron-to- SC negative ion temperature ratio (σn ) change. 1 1 M AN U Keywords: Electronegative Plasmas; Modulational Instability; Modulation angle. Introduction In the last twenty years, solitonic structures have been studied in a broad range physical 3 systems, especially in nonlinear optics [1, 2, 3, 4], biophysics [5, 6, 7, 8] and in Bose-Einstein 4 condensates [9, 10]. They originate from the competition between nonlinear and dispersive 5 effects, and can move over long distances, with unaltered characteristics. The physics of dusty 6 plasmas has significantly been related to linear and nonlinear waves, usually observed in space 7 and laboratory plasmas. For example, ion-acoustic solitons have been the object of intense 8 investigations, both theoretically and experimentally in plasmas comprising electrons and posi- 9 tive ions [11, 13, 14, 15, 16, 17]. In electronegative plasmas (ENPs), one finds both negative and 10 positive ion species, as well as electrons. Negative ion plasmas may be obtained as a result of 11 basic processes, such as dissociative or non-dissociative electron attachment to neutrals, espe- 12 cially when electronegative gases are injected into an electrical gas discharge or injected from an 13 external source [11, 12, 18]. In ENPs, there should be a critical concentration of negative ions 14 for compressive soliton propagation to be possible. Indubitably, this is straightforwardly related 15 to the critical negative ion density, which for some values beyond that critical one, may support 16 the emergence of rarefractive solitons [19]. Nevertheless, if at the critical density, the interplay 17 between dispersive and nonlinear effects is lost and consequently the usual Korteweg-de Vries 18 (KdV) theory will no more be suitable for studying soliton propagation [20, 21]. However, a AC C EP TE D 2 2 ACCEPTED MANUSCRIPT time-dependent perturbation may lead to a modified KdV (mKdV) equation, only in the pres- 20 ence of higher-order nonlinearity [22]. More recently, experimental observations of Peregrine 21 solitons in nonlinear optical fibers [23, 24], water tank experiment [25, 26] and in plasmas [27, 28] 22 have opened a new route to study their characteristics more deeply. Along the same lines, Ion 23 acoustic waves (IAWs) are found to be modulationally unstable when the plasma contains a 24 critical amount of negative ions and described by the nonlinear Schrödinger (NLS) equation. 25 Mamun et al. [11], based on laboratory experiments, have recently paid attention to the ex- 26 istence of IA and DIA waves in an electronegative plasma made of Boltzmann negative ions, 27 Boltzmann electrons and cold mobile positive ions. Also, we have highlighted, in our previ- 28 ous works, the impact of the negative ion concentration ratio and the electron-to-negative ion 29 temperature ratio on the modulational instability (MI) both in one- and two dimensional con- 30 texts [29, 30]. In fact, the MI of nonlinear excitations in plasmas is a well-known phenomenon 31 leading to energy localization, the main consequence being the formation of bright envelope 32 solitons. This means that, in the absence of instability, dark solitons are the most probable 33 excitations to emerge in such systems. MI therefore originates from the fact that a small plane 34 wave perturbation grows exponentially and the resulting sidebands get amplified, to finally dis- 35 play trains of oscillations. In general, the subsequent bright solitons are solutions of the NLS 36 equation, which can be derived from generic hydrodynamic plasma equations using appropriate 37 expansion methods such as the reductive perturbative method [16, 17], the derivative expansion 38 method [31], the Kryslov-Bogoliubov method [32, 33], to name just a few. In more recent contri- 39 butions, particular attention has been paid to the multi-dimensional versions of such methods, 40 leading to more upgraded amplitude equations such as the Davey-Stewartson (DS) equation 41 and the multi-dimensional NLS equation, with at least two space variables. The MI and soliton 42 solutions of the 2D-DS equations have been recently addressed, with emphasis on the effects 43 of the ENP parameters. Periodic solutions and MI of the DS were also proposed by Tajiri et 44 al. [34]. Gill and co-workers [35] also studied 2D envelope electron acoustic waves in the pres- 45 ence of Cairns non-thermal distribution of hot electrons. Bedi and Gill [15] studied envelope 46 electron acoustic waves subjected to transverse perturbations, in the presence of κ−distributed 47 hot electrons. In three dimensions, Carbonaro [14] derived DS equations from a plasma system 48 consisting of cold electrons, hot electrons and steady background of ions, and further supported 49 the idea of Kourakis and Shukla [13] that in higher dimensions the MI phenomenon is mostly 50 controlled by the modulation angle, leading to parallel, transverse and oblique modulations. 51 The concept is also introduced in the present work and applied to ENPs. We study the dynam- 52 ical outcomes of the interplay between ENP parameters and the angle of modulation using the AC C EP TE D M AN U SC RI PT 19 3 ACCEPTED MANUSCRIPT 53 MI technique. The 3D-DS equations are first derived via the reductive perturbation method, 54 followed by a comprehensive parametric linear stability analysis of plane wave solutions. Some 55 particular cases such as the parallel and transverse modulations are discussed. Some concluding 56 remarks ends the paper. 57 2 58 We consider an unmagnetized electronegative plasma system composed of Maxwellian electrons 59 and negative ions in addition to cold mobile positive ions [11, 29, 36]. The nonlinear features 60 of the IAWs may be described by the following set of coupled normalized ion-fluid equations in 61 three-space dimension: (1b) ∆φ = µe exp φ + µn exp σn φ − ni , (1c) 63 65 where ni is the number density of positive ions, which is normalized by the unperturbed value → − − − − ni0 . V = u→ ex + v → ey + w → ez , where u, v and w are the velocities of charged dusts (with TE D 64 (1a) → − → −−→ − − −−→ → ∂V + V · grad V = −gradφ, ∂t M AN U 62 SC  → − ∂ni + div ni V = 0, ∂t RI PT Mathematical Model 66 mass mi ) in x, y and z directions, respectively. The overall charge neutrality at equilibrium 67 is ni (0) 69 70 (0) EP 68 (0) = ne + nn . The different variables that appear in Eqs. (1a)-(1c) have also been → − adequatly normalized: ni is normalized by the unperturbed ion density ni0 ; V is normalized p by the dust-acoustic (DA) speed c = ZkB Te /mi , with Te denoting the electron temperature, kB the Boltzmann constant and Z the charged dust state, i.e., the number of electrons per ion found on the dust-grain surface. φ represents the electrostatic wave potential and is normalized 72 by kB Te /e, where e is the magnitude of the electron charge. The time and space variables 73 are normalized by the ion-Debye length λD = (kB T e/4πe2 ni ) 74 ω −1 = (4πe2 ni0 /mi )−1/2 , respectively. σn = Te /Tn is the electrons-to-negative ion temperature 75 ratio, µe = ne0 /ni0 and µn = nn0 /ni0 , where ni0 , nn0 and ne0 , are the unperturbed densities 76 of the positive ions, negative ions and electrons, respectively. At equilibrium, the neutrality 77 condition of the plasma reads µe + µn = 1, where µe = ne0 /ni0 = 1/(1 + α), with α = nn0 /ne0 . 78 Using the power series expansion of the exponential functions around zero, Eq. (1c) becomes AC C 71 1/2 and the ion plasma period ∂ 2φ ∂ 2φ ∂ 2φ + 2 + 2 = 1 + a1 φ + a2 φ2 + a3 φ3 − ni , 2 ∂x ∂y ∂z 79 where, a1 = µe + µn σn , a2 = 2 µe +µn σn 2 and a3 = 3 µe +µn σn . 6 4 (2) ACCEPTED MANUSCRIPT 80 In order to investigate the propagation of IAWs and derive the amplitude equations for the 81 above-described plasma system, we employ the standard reductive-perturbation technique. We 82 introduce the stretched variables in space and time as, ξ = ǫ(x − vg t), η = ǫy, ζ = ǫz and τ = ǫ2 t, where the group velocity vg will be determined later by the solvability condition of 84 Eqs. (1). The dependent physical variables around their equilibrium values are given by the 85 trial expressions n=1+ ∞ X ǫ p p=1 +∞ X (p) nil (ξ, η, ζ, τ ) Al (x, t) l=−∞ 86 φ= ∞ X p=1 ǫp +∞ X (p) φl (ξ, η, ζ, τ ) Al (x, t) l=−∞  (p) ul (ξ, η, ζ, τ ) → −  l  (p) ǫp V =  vl (ξ, η, ζ, τ )  A (x, t). (p) p=1 l=−∞ wl (ξ, η, ζ, τ ) +∞ X  (3a) (3b) (3c) M AN U ∞ X 88 SC 87 RI PT 83 We Note that the above series include all overtones Al (x, t) = exp[il(kx − ωt)] up to order p, 92 generated by the nonlinear terms, i.e., the corresponding coefficients are of maximum order ǫp . ∗  ∗  (p) (p) (p) (p) = u−l , = ni−l , ul The reality condition of physical variables requires the relations nil ∗ ∗  ∗   (p) (p) (p) (p) (p) (p) vl = v−l , wl = w−l and φl = φ−l to be satisfied. The asterisk denotes the 93 (2), and equating the quantities with equal power of ǫ, we obtain at ǫ1 order, for l = 1, the 94 following solutions corresponding to the first harmonic of perturbation 90 91 complex conjugation. Substituting trial solutions (3a)-(3c) into the basic Eqs. (1a), (1b) and (1) 1 ω (1) (1) (1) ni1 , u1 = ni1 , v11 = 0, w11 = 0, + a1 k (4) given that the dispersion relation k2 ω = 2 k + a1 2 AC C 95 k2 EP φ1 = TE D 89 (5) 96 be satisfied. We process the same way and obtain the second-order terms, namely the ampli- 97 tudes of the second harmonics and constant terms as well as the non vanishing contributions 98 to the first harmonics. We obtain the following equations at O(ǫ2 )−order, for l = 0, (2) (2) (1) a1 φ0 − ni0 − 2a2 |φ1 |2 , 99 (6) and (1) − vg (1) (1) (1) ∂ni1 ∂u ∂u ∂φ (2) (2) (2) (2) − iωni1 + iku1 + 1 = 0; −vg 1 − iωu1 + ikφ1 = − 1 = 0; ∂ξ ∂ξ ∂ξ ∂ξ (1) (2) − iωv1 (1) (1) ∂φ ∂φ ∂φ (2) (2) (2) = − 1 , −iωw1 = − 1 ; (k 2 + a1 )φ1 − ni1 = 2ik 1 ∂η ∂ζ ∂ξ 5 (7) ACCEPTED MANUSCRIPT 100 for l = 1. For l = 2, the system reduces to (2) (2) (1) (1) (1) (1) (1) (2) − 2iωni2 + 2iku2 + 2ikni1 u1 = 0; −2iωu2 + ik(u1 )2 + 2ikni1 u1 = 0; 2 (4k + 101 (2) a1 )φ2 − (2) ni2 + (1) a2 (φ1 )2 = 0; (2) −2iωv2 = 0, (2) −2iωw2 (8) = 0, which provides the compatibility condition ω3 . k3 (9) RI PT vg = a1 (2) (2) (2) 102 By solving equation (8), we find the second harmonic quantities n2i , u2 and φ2 in term of 103 φ1 in the form (1) with αφ = 1 ω a2 a2 − 2 2 , αn = (a1 + 4k 2 )αφ + (αn − 1). 2 , αu = 2 2 2 2k 3k (k + a1 ) k (k + a1 ) M AN U 104 (10) SC 2 2   2  (2) (2) (1) (1) (2) (2) (1) (2) , v2 = w2 = 0, , u2 = αu ni1 , ni2 = αn ni1 φ2 = αφ ni1 (11) Moreover, the expression for the zeroth harmonic mode cannot be determined completely within 106 the second order, so we will have to consider the third-order equations. Therefore, the set of 107 equations given by the (l = 0)−components of the third-order part of the reduced equations is 108 given by (2) (2) (2) (2) TE D 105 (2) (2) (1) (2) (1) (2) ∂u ∂v ∂w0 2ω ∂|ni1 |2 ∂v ∂φ ω 2 ∂|ni1 |2 ∂n + = 0; −vg 0 + 0 + 2 = 0; − vg i0 + 0 + 0 + ∂ξ ∂ξ ∂η ∂ζ k ∂ξ ∂ξ ∂ζ k ∂ζ ∂ 2 φ2 δ1 20 − ∂ξ 110 (2) (1) (2) (12) to which we add Eq. (6) from O(ǫ2 ), for l = 0. From Eqs. (12), we get AC C 109 (1) ∂φ ω 2 ∂|ni1 |2 ∂v ∂φ ω 2 ∂|ni1 |2 ∂u0 + 0 + 2 = 0; −vg 0 + 0 + 2 = 0, ∂ξ ∂ξ k ∂ξ ∂ξ ∂η k ∂η EP − vg with  ∂ 2 φ20 ∂ 2 φ20 + ∂η 2 ∂ζ 2 δ1 = vg2 a1  (1) ∂ 2 |ni1 |2 − δ2 − δ3 ∂ξ 2 (1) (1) ∂ 2 |ni1 |2 ∂ 2 |ni1 |2 − ∂η 2 ∂ζ 2 ! = 0, ω2 2vg ω ω 2 2a2 vg2 ω 4 , δ = . + 2 − − 1, δ2 = 3 k k k4 k2 (13) (14) 111 The various expressions found in the above calculations are then introduced into the (l = 112 1)−component of the third-order part of the equations. This leads to the following amplitude 113 equation i (1) ∂ni1 ∂τ + (1) ∂ 2 ni1 γ1 ∂ξ 2 + γ2 (1) ∂ 2 ni1 ∂η 2 + (1) ∂ 2 ni1 ∂ζ 2 6 ! (1) (1) (2) (1) + γ3 |ni1 |2 ni1 + γ4 φ0 ni1 = 0, (15) k = 0.6; n ACCEPTED MANUSCRIPT k = 0.6; = 16.0 = 5.5 =0.01 = 0.08 = 0.2 k = 0.6; n 10 6 = 0.01 = 0.08 = 0.2 8 4 n = 22.5 = 0.01 = 0.08 = 0.2 20 15 10 2 4 5 2 0 0 0 -2 -5 -2 (a) -4 0 0.2 0.4 (b) -4 0.6 0.8 1 0 0.2 0.4 / (c) -10 0.6 0.8 1 / 0 0.2 0.4 0.6 RI PT P Q 6 0.8 1 / 114 SC Figure 1: The panels show the plots of the product P × Q versus the modulation angle θ, under the influence of the plasma parameters σn and α. Each panel corresponds to a value of σn submitted to the increasing effect of α. where 2(k 2 γ4 = − M AN U a1 −3ka1 , γ2 = , 5 2 + a1 ) /2 2k(k + a1 )3 /2 " # k 4a2 2a2 2a22 3a3 2a2 2k 2 3a1 γ3 = − + + − + 2− 2 2 − 6+ , 2(k 2 + a1 )1/2 a1 2k 3k (k + 1) 3k 2 (k 2 + a1 )2 3k 2 (k 2 + a1 )3 (k 2 + a1 )3 γ1 = k(k 2 + a1 )3/2 ka2 ka1 + 2 . − 2 1/2 a1 2(k + a1 ) (k + a1 )3/2 (1) (16) (2) Further introducing the notations F = ni1 and G = φ0 , the coupled equations (13) and (15) 116 become TE D 115 ∂F ∂ 2F i + γ1 2 + γ2 ∂τ ∂ξ 117  ∂ 2G ∂ 2G + ∂η 2 ∂ζ 2 EP ∂ 2G δ1 2 − ∂ξ  ∂ 2F ∂ 2F + ∂η 2 ∂ζ 2  ∂ 2 |F |2 − δ2 − δ3 ∂ξ 2  + γ3 |F |2 F + γ4 GF = 0,  ∂ 2 |F |2 ∂ 2 |F |2 + ∂η 2 ∂ζ 2  = 0. (17a) (17b) The above system (17) represents the DS equations that were initially derived to describe mod- 119 ulated waves packets in water of finite depth [37]. Their soliton solutions were then investigated 120 via the inverse scattering transform [38]. Dromion solutions in the 2D context were also derived 121 recently with emphasis on their interaction and energy exchange [30]. 122 3 123 The DS Eqs. (17) admit the trivial homogeneous solutions F = F0 eiγ3 F0 τ and G = 0, where F0 is 124 a real constant that represents the amplitude of carrier wave. MI of IAWs is investigated under 125 small perturbations in phase, in amplitude or in both. Since we are interested in amplitude 126 modulation, the corresponding perturbed solutions then write F = (F0 + δF (ξ, η, ζ, τ ))eiγ3 F0 τ AC C 118 Modulational instability 2 2 127 and G = δG(ξ, η, ζ, τ )) with δF ≪ F0 . After linearizing Eqs. (17) around the unperturbed 7 ACCEPTED MANUSCRIPT 128 plane wave solutions, we obtain the governing equations for the small perturbations δF and δG 129 in the form ∂ 2 δF ∂δF + γ2 + γ1 i ∂τ ∂ξ 2  ∂ 2 δF ∂ 2 δF + ∂η 2 ∂ζ 2  + γ3 F02 (δF + δF ∗ ) + γ4 δGF0 = 0, (18a) 130 131  ∂ 2 δG ∂ 2 δG + ∂η 2 ∂ζ 2  ∂ 2 (δF + δF ∗ ) ∂ξ 2  2  ∂ (δF + δF ∗ ) ∂ 2 (δF + δF ∗ ) − δ3 F0 = 0. + ∂η 2 ∂ζ 2 − δ2 F0 RI PT ∂ 2 δG − δ1 ∂ξ 2 (18b) We make use of the transformation δF = a + ib and δG = c + id, with 134 135 M AN U 133 and obtain the nonlinear dispersion relation     2  2F02 δ2 γ4 µ21 + δ3 γ4 (µ22 + µ23 ) 2 2 2 2 − γ3 . (19) 1+ Ω = γ1 µ1 + γ2 (µ2 + µ3 ) (γ1 µ21 + γ2 (µ22 + µ23 )) −δ1 µ21 + µ22 + µ23 The perturbation wavenumber vector can be expressed using spherical coordinates, i.e., (µ1 , µ2 , µ3 ) = (K cos θ, K sin θ cos ϕ, K sin θ sin ϕ). Eq. (19) then reduces to   2 2Q 2 2 2 Ω = K P K − 2F0 , P where TE D 132 SC (a, b, c, d) = (a0 , b0 , c0 , d0 )ei(µ1 ξ+µ2 η+µ3 ζ−Ωτ ) , δ2 cos2 θ + δ3 sin2 θ P = γ1 cos θ + γ2 sin θ and Q = γ3 + γ4 . δ1 cos2 θ − sin2 θ 137 138 139 140 (21) There will be instability if the frequency Ω is complex, i.e., Ω2 < 0. According to expression (20), this mainly depends on the product P × Q and the value of the perturbation wavenumber q . Although we obtain a result similar to the one in which is such that K < Kcr = F0 2Q P AC C 136 2 EP 2 (20) Ref. [13], we remark here that the instability condition depends of the angle θ which may lead to different instability scenarios as shown in Fig. 1, where P × Q is plotted versus the 141 modulation angle θ, additionally to the effects of the plasma parameters. In Fig. 1(a), for 142 example, σn = 5.5 and one observes two lateral regions of instability for α = 0.01, especially 143 in the intervals 0.1π ≤ θ ≤ 0.23π and 0.75π ≤ θ ≤ 0.88π. However, with α = 0.08 and 0.2, 144 one observes a central region of instability which excludes the lateral ones observed previously. 145 This later behavior persists for σn = 16 as the central instability interval of θ gets expanded as 146 α increases (see Fig. 1(b)). More interestingly, lateral regions of instability appear once more 147 for σn = 22.5, with α = 0.01, in the intervals 0 < θ ≤ 0.22π and 0.78π ≤ θ ≤ π (see Fig. 1(c)). 8 TE D M AN U SC RI PT ACCEPTED MANUSCRIPT 148 149 AC C EP Figure 2: The growth rate of MI is plotted versus the wavenumber K and the electron-tonegative ion temperature ratio σn in the generalized case, i.e., θ 6= 0. Panels (aj)j=1,2,3 corresponds to to θ = π/10, panels (bj)j=1,2,3 , gives results for θ = π/5 and panels (cj)j=1,2,3 have been recorded for θ = π/3. The three columns correspond to different values of the wavenumber k, with α = 0.8. With increasing α, the previous central region where P × Q > 0 appears again, and tends to expand. In general, MI is characterized by its growth rate given by the expression r √ QF02 Γ = −Ω2 = |P K| 2 − K 2. P (22) 150 Fig. 2 is a good illustration of the above growth rate of instability which has been plotted 151 versus the perturbation wavenumber K and the electron-to-negative ion temperature ratio σn . 152 We have in fact considered different values of the modulation angle θ to clearly illustrate what 153 is discussed in Fig. 1. Panels (aj)j=1,2,3 have been plotted for θ = π/10, a value that gives 154 rise to instability domains. Especially, for k = 0.70, one notices the coexistence of two regions 155 of instability both for very small and high σn , that disappear with increasing θ as shown in 9 ACCEPTED MANUSCRIPT 50 0 8 6 40 -0.1 4 30 -0.2 2 P Q P Q 20 10 -0.3 0 -2 -4 0 -6 -0.4 -10 -0.5 -8 (a2) -20 0 0.5 1 1.5 2 0 0.2 n 10 0.4 = 5.0 0.6 n 0.8 = 16.0 1 n 0 = 22.5 20 30 (b1) (b2) 8 20 6 10 15 0 2 -10 5 0.2 0.4 0.6 0.8 1 (b3) SC P Q P Q 10 4 (a3) -10 RI PT (a1) 0 -5 0 -20 0 0.2 0.4 0.6 0.8 1 k 0 M AN U -10 0.2 0.4 0.6 0.8 1 k -15 0 0.2 0.4 0.6 0.8 1 k TE D Figure 3: The dispersion coefficient P , the nonlinearity coefficient Q and the product P × Q are depicted versus the wavenumber k for different values of the electron-to-negative ion temperature ratio σn . Panels (aj)j=1,2,3 correspond to the parallel modulation, i.e., θ = 0, while panels (bj)j=1,2,3 stand for the perpendicular modulation, i.e., θ = π/2. The solid blue line corresponds to σn = 5, the dashed-red line corresponds to σn = 16.0 and the dotted-yellow line corresponds to σn = 22.5, with α = 0. Fig. 2(b3). For the rest, θ and k have the effect of reducing the instability domain expansion. 157 Those regions are where modulated IAWs are expected, depending on the right choice of both 158 the wave and plasma parameters. 159 EP 156 In what follows, depending on the value of θ, we address two main cases known as the parallel and the transverse modulations [13]. 161 A) Parallel modulation 162 Parallel modulation is obtained for θ = 0, which reduces the coefficients P and Q to the 163 simplified expressions AC C 160 P = γ1 and Q = γ3 + γ 4 δ2 δ1 (23) 164 Interestingly, the above two coefficients still depend of the wavenumber k as clearly depicted by 165 Fig. 3 (a1) and (a2). In this case, the sign of P remains negative with changing the value of the 166 electron-to-negative ion temperature ratio σn . However, due to the later, there are regions of k 10 M AN U SC RI PT ACCEPTED MANUSCRIPT 167 168 169 TE D Figure 4: The growth rate of MI is plotted versus the wavenumber K and the electron-tonegative ion temperature ratio σn for the parallel modulation (θ = 0). The columns, from left to right correspond respectively to k = 0.1, 0.18 and 0.20. The upper line, i.e., panels (aj)j=1,2,3 corresponds to to α = 0.1 and the lower line, made of panels (bj)j=1,2,3 , gives results for α = 0.5. where the nonlinearity coefficient Q is positive or negative. This brings about some instability regions as shown in Fig. 3(a3), where we have plotted the product P × Q. Specifically, for σn = 5, P × Q presents two positive regions, i.e., 0 < k < 0.25 and 0.32 < k < 0.85. For the rest, only one region of instability exist for σn = 16 and 22.5, which are respectively 171 0.3 < k < 0.5 and 0.42 < k < 0.6. Using these values of the wavenumber k, we have also 172 plotted the MI growth rate in Fig. 4 versus the perturbation wavenumber K and σn . While 173 the different panels (aj)j=1,2,3 correspond to different values of the wavenumber k, they have 174 been plotted when the negative ion concentration ratio takes the value α = 0.1. There, regions 175 of instability are detected and one sees how sensitive they are to the values of k. Along the 176 same line, still considering the previous values of k, the MI growth rate has been plotted for 177 α = 0.5 and the corresponding results are recorded in Fig. 4(bj)j=1,2,3 . Compared to the case in 178 panels (aj)j=1,2,3 , regions of instability are restricted to small values of the electron-to-negative 179 ion temperature ratio σn . AC C EP 170 11 M AN U SC RI PT ACCEPTED MANUSCRIPT TE D Figure 5: The growth rate of MI is plotted versus the wavenumber K and the electron-tonegative ion temperature ratio σn for the perpendicular modulation (θ = π/2). The columns, from left to right correspond respectively to k = 0.22, 0.38 and 0.70. The upper line, i.e., panels (aj)j=1,2,3 corresponds to to α = 0.1 and the lower line, made of panels (bj)j=1,2,3 , gives results for α = 0.5. B) Transverse modulation 181 The transverse modulation is obtained for θ = π/2, which reduces P and Q to EP 180 P = γ2 and Q = γ3 − γ4 δ3 . (24) They are plotted in Figs. 3 (b1) and (b2). Contrarily to the case θ = 0, the dispersion parameter 183 P remains positive with changing σn , while the value of Q is very sensitive to such a change. 184 AC C 182 Also in this case, there are regions of instability, i.e., where P × Q > 0, as shown in Fig. 3(b3). 185 Obviously, regions of k where MI is expected are the opposite of what has been obtained in 186 Fig. 3(a3). If only very limited values of k can give rise to instability for σn = 5, regions of 187 instability are more obvious for σn = 16 and 22.5. For each of the later cases, there are two 188 regions of instability 0.23 < k < 0.5 and 0.5 < k < 0.55 for σn = 16; 0.45 < k < 0.6 and 189 0.6 < k < 0.75 for σn = 22.5. The detected regions of k may indeed give rise to unstable 190 patterns as shown in contour plot of the MI growth rate of Fig. 5. In Figs. 5(aj)j=1,2,3 , we have 191 considered α = 0.1 as in Fig. 4, except that here, one notices a delocalization of the instability 192 domain in the (σn , K)−plane. To remind, for the case of the parallel modulation, instability 12 ACCEPTED MANUSCRIPT 193 has been detected in regions of small σn , i.e., values that belong to the interval 0 < σn < 10. 194 In this case, i.e., the transverse modulation, modulated IAWs may mainly be found in regions 195 belonging to the interval 10 ≤ σn ≤ 30. In Fig. 5(bj)j=1,2,3 , we have fixed α = 0.5. Compared 196 to Fig. 4(a1), the regions of K giving rise to instability are larger in Fig. 5(b1). There is 197 indeed instability delocalization for k = 0.38, where regions of instability belong to the interval 199 9.8 ≤ σn ≤ 28.5 (see Fig. 5(b2)). The detected region gets more reduced both in the K− and σn −directions when k = 0.70 as depicted in Fig. 5(b3). RI PT 198 4 Conclusion 201 In the present paper, we have addressed the MI of IAWs in a 3D unmagnetized electronegative 202 plasma model. Using the reductive-perturbation approach, we have shown that the model 203 equations can be reduced to a set of DS equations in three dimensions, whose coefficients have 204 been found to be dependent on the plasma parameters. Under the activation of MI, we have 205 detected various dynamical modes, this because of the presence of the modulation angle θ. 206 In that respect, a comprehensive parametric analysis of wave instability has been conducted, 207 where regions of instability/stability have been revealed to be very sensitive to θ, σn and α. Two 208 main cases have finally been addressed, the longitudinal (θ = 0) and the transverse (θ = π/2) 209 modulations. It has in general been found that the two regimes do not belong to the same 210 intervals of the electron-to-negative ion temperature ratio (σn ) and the wavenumber k. This 211 indeed shows that they have different dynamical features mainly under the activation of MI. 212 The later can support a broad range of excitations depending on the values of the original 213 parameter like α and σn . When the used parameters fall well inside the instability regions, ion 214 acoustic solitons may be expected to emerge and display coherent spatiotemporal behaviors. 215 Also, the strong relationship between IAWs and MI has been discussed intensively, including 216 the emergence of exotic solitons like dromions and rogue waves. In the proposed model, finding 217 such exact solutions remains an opened problem which is actually under investigation, especially 218 in the presence of magnetic and relativistic effects. 219 Acknowledgements 220 The work by CBT is supported by the Botswana International University of Science and Tech- 221 nology under the grant DVC/RDI/2/1/16I (25). AC C EP TE D M AN U SC 200 13 222 223 224 ACCEPTED MANUSCRIPT References [1] D. 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