Nonlinear propagation of modulated ion-acoustic plasma waves in the presence of an electron beam

Transcription

1 1 PoP29447 Nonlinear propagation of modulated ion-acoustic plasma waves in the presence of an electron beam A. Esfandyari-Kalejahi a, I. Kourakis b, B. Dasmalchi a and M. Sayarizadeh a a: Azarbaijan University of Tarbiat Moallem, Faculty of Science, Department of Physics, , Tabriz, Iran. b: Institut fuertheoretische Physik IV, Fakultaet fuerphysik und Astronomie, Ruhr-Universitaet Bochum, D Bochum, Germany Submitted 13 October 2005, revised 8 February 2006 Abstract Theoretical and numerical studies are presented of the amplitude modulation of ion-acoustic waves (IAWs) in a plasma consisting of warm ions and a cold electron beam. Perturbations parallel to the carrier IAW propagation direction have been investigated. The existence of four distinct linear ion acoustic modes is shown, each of which possesses a different behavior from the modulational stability point of view. The stability analysis, based on a nonlinear Schroedinger equation (NLSE) reveals that the IAW may become unstable. The stability criteria depend on the IAW carrier wave number, and also on the ion temperature, the beam velocity and the beam electron density. Furthemore, the occurrence of localized envelope structures (solitons) is investigated, from first principles. The numerical analysis shows that the two first modes (essentially IAWs, modified due to the beam) present a complex behaviour, essentially characterized by modulational stability for large wavelengths and instability for shorter ones. Dark-type envelope excitations (voids, holes) occur in the former case, while bright-type ones (pulses) appear in the latter. The latter two modes are characterized by an intrinsic instability, as the frequency develops a finite imaginary part for small ionic temperature values. At intermediate temperatures, both bright- and dark-type excitations may exist, although the numerical landscape is intertwined between stability and instability regions. PACS Nos. Keywords: Fp, Mw, Sb. Ion-acoustic waves, modulational instability, beam plasma instability, Nonlinear Schroedinger Equation, envelope solitons.

2 2 I. Introduction Electrostatic (ES) and/or electromagnetic (EM) plasma wave related solitary structure formation and propagation is a long-standing problem, which has been gathering significant interest among plasma physicists, in particular since spacecraft missions have enabled in-situ measurements of electric and magnetic fields e.g. in the magnetosphere and beyond [1]. Abundant space observations indicate that stationary nonlinear ion-acoustic structures occur widely in space. Such localized ES structures may, for instance, be responsible for the acceleration of auroral particles to kv energies [2]. A number of localized ES excitations traced e.g. by the GEOTAIL mission have been reported to propagate at speeds much slower than the electron thermal speed, and have thus been effectively interpreted as ion-acoustic (IA) wave related nonlinear excitations [3], while subsequent theoretical [4] and ab-initio numerical investigations have confirmed those results [5, 6]. Of particular interest is the case when an electron beam is present in a plasma. Such a situation is typically encountered in the upper layers of the magnetosphere, where the co-existence of two different electron populations (say, cold i.e. inertial and warm i.e. energetic ones) has been reported by satellite missions, e.g. the FAST at the auroral region [7,8], the S3-3 [9], Viking [10], GEOTAIL and POLAR [8, 11] missions. Such a beam-plasma system has also been created in laboratory [12, 13, 14], where modulated envelope wave packets our focus issue here were created and observed. From a theoretical point of view, the existence of an electron beam in the plasma has been shown to modify the properties and conditions for existence of arbitrary amplitude localized ES nonlinear excitations [15-17] (typically modelled via the Sagdeev pseudopotential formalism), in addition to small amplitude solitary pulses [18-20] (i.e. generically related to Korteweg-deVries and related equation theories, derived via reductive perturbation techniques). Such theoretical considerations have later been extended to (higher-frequency) electron-acoustic waves [21] (related to Broadband Electrostatic Noise, BEN, in the Earth's auroral region), supported by numerical simulations [22], and also to ion-beam effects [23]. Interestingly, an extended dimension was offered to the beam-plasma problem by a recent pair (electronpositron) beam-plasma experiment, in which transmitting a low-energy electron beam through positron plasmas stored in a Penning trap was shown to lead to a large amplitude oscillations of the positron plasma and even positron ejection [24].

3 3 One of the generic nonlinear paradigms involved in the description of wave behavior in plasmas is the nonlinear Schroedinger (NLS) equation [25, 26], which generically models the dynamics of (the amplitude of) a nonlinearly modulated envelope wave packet. In addition to extended quasi-periodic solutions, the (integrable) NLS Equation possesses stationary profile envelope solutions, i.e. localized structures (envelope solitons) bearing the form of a localized envelope excitation which confines (modulates, in space) a fast internal ES carrier wave oscillation. Indeed, localized envelope solitary structures accompanied by localized density depletions have often been observed by rocket missions in the ionosphere and the magnetosphere [27, 28] (also for EM modes [29]). From first theoretical principles, soliton solutions of the NLS Eq. may be either of the bright type (where a pulse-shaped, slowly vibrating localized form, with vanishing boundaries, modulates the internal carrier wave), or of the dark type (where a negative pulse-shaped localized disturbance, i.e. a propagating ES potential dip, or hole, occurs in an otherwise constant amplitude, harmonic environment). Following standard terminology, the latter may be of the black or gray type, depending on whether the potential hole reaches a zero or a finite, respectively, value in the center of the excitation. A detailed analysis of the topology and properties of these soliton solutions of the NLS Eq. can be found in the bibliography (see e.g. Refs 30; also for a summary, with respect to ES plasma modes [34-35]), so relevant information provided here will be kept to a minimum detail. Apart from modeling nonlinear plasma waves, envelope structures occur in a wide range of physical contexts, ranging from nonlinear optics (pulse propagation in optical fibers) to solid state physics, transmission line experiments and charge transport in biological macromolecules [36]. In a generic fashion, the transition from a harmonic (linear) behavior to a nonlinear multi-soliton is believed to be related to the mechanism of modulational instability (MI), which describes a breakdown of the amplitude of a harmonic wave due to nonlinearity. This nonlinear effect is first manifested as a superposition of linear harmonics of the carrier phase, and may then evolve beyond the weak perturbation range until final wave collapse, unless structuring (in the form of soliton formation) occurs, thanks to a balance between nonlinearity and wave group dispersion. With respect to ES plasma modes [34-35], MI has been been studied via different fluid plasma models [31-33, 37, 38]. Early experiments have indeed shown this tendency for energy localization via MI [39], and numerical simulations (e.g. on EM modes [40]) also predict such a behavior. Despite the obvious importance of MI and related stable envelope structure formation in stable ES wave packet propagation, a relevant study in electron-beam plasma systems is still lacking in literature, to our knowledge. This work is devoted in a detailed study of such effects,

4 4 from first principles, in unmagnetized plasmas characterized by the presence of an electron beam. The layout of this manuscript is as follows. In the next section, the analytical model is introduced and then employed, in Section III, as the basic of a perturbative analysis by introducing appropriate slow space and time evolution scales. An NLS-type equation is derived, governing the (slow) amplitude evolution in time and space. Exact expressions for soliton solutions of the NLSE are provided. Section IV is devoted to a discussion of the linear stability of IA waves by means of numerical calculations. Finally, section V is devoted to summarizing our conclusions. II.The model equations The basic set of fluid equations for ions and the cold electron beam read: n i t x n i u i =0, (1) u i t +u u i i x φ x n b t 3σ 1+α n 2 i n i x =0, (2) x n b u b =0, (3) t u b +u b The system is closed by Poisson's equation φ 2 u b x 1 μ φ =0. (4) x x 2 =n e +n b n i, (5) where a Boltzmann-type distribution is considered for background electrons,, hence n e =e φ, given the low ion acoustic wave frequency range we are interested in. Eqs. (1) (5) constitute the closed set of equations which models our problem. Here σ=t i /T e, α=n b0 /n 0 and μ=m e /m i where T i ( T e ), n 0 ( n bo ) and m i ( m e ) are the ion (electron) temperature, the unperturbed background electron (electron beam) density, and the ion (electron) mass, respectively. The variables n e, n b, n i, u b, u i and φ denote the background electron density, the electron beam electron density, the ion density, the beam electron velocity, the ion velocity, and the electrostatic potential, respectively. Velocities are normalized by the ion sound speed V s = K B T e /m i 1/2. Furthermore, we have scaled the time t and space x variables by the (inverse) ion plasma frequency ω pi = n 0 e 2 /ε 0 m i 1/2 and the electron Debye length λ d = ε 0 K B T e /n 0 e 2 1/2, respectively; finally, densities are scaled by the background electron density n e,0 and the potential is scaled by

5 5 K B T e /e (where e is the electron charge). Overall charge neutrality is assumed at equilibrium, implying n e,0 +n b,0 n i,0 =0 (or n i,0 =1+α ). The electron beam density is assumed to be small, as compared to the bulk plasma ion density, thus ensuring overall current quasi-neutrality. III.Methodology derivation of the modulated amplitude evolution equation (i) The perturbative analysis In order to obtain an explicit evolution equation for the modulating IAW envelope, from the basic model Eqs. (1) - (5), we shall employ the standard reductive perturbation (multiple scale) technique [41]. We consider the stretched space and time variables ξ=ε x λt and τ=ε 2 t, where ε <<1 is a small real parameter and λ is an arbitrary real parameter, to be later determined by compatibility requirements. The dependent variables n i, u i, n b, u b and φ constitute the five components S l n ξ,τ of a dynamical state vector, say S, which is here expanded near the equilibrium state S 0 0 ={1+α,0,α,u b,0,0 } as S l =S l n=1 ε n n S l ξ,τ exp [il kx ωt ]. The reality condition l= S n n l =S l is met by all state variable amplitudes; the star superscript denotes complex conjugate. Substituting the expansion above ansatz and isolating distinct orders in ε, a set of (5) nth order reduced equations are obtained. These equations, reported in A.1 in the Appendix for clarity, determine the amplitudes S l n ξ,τ of the lth harmonic contributions of each state variable S l, to order n in ε. The first-order (n=1) equations read ilωn l 1 1+α ilk u l 1 =0, 1 ilωu l +ilk φ 1 l 3σ 1+α ilkn 1 l =0, 1 ilωn b,l +ilαk u 1 b,l +ilku b0 n 1 b,l =0, (6) 1 ilωu b,l +ilk u b0 u 1 b,1 1 μ ilkφ 1 l =0,

6 6 l 2 k 2 φ 1 1 φ 1 l n 1 bl +n 1 l =0 from which the following dispersion relation for l=1 is deduced: 1+α ω 2 k 3 σ 2 α μ u b0 ω k 2 =1+k 2. (7) Expanding Eq. (7), a 4 th order polynomial in the carrier frequency ω is obtained. The ordinary IAW dispersion curve ω=± k 2 1+k 3 σk 2 =±ω 2 IAW (valid in e-i plasmas) is thus found to split to four distinct modes, two of which are only due to the presence of the beam (and vanish in its absence), while the two remaining ones are essentially modified ion acoustic modes (distorted due to the beam). The former two branches are in fact unstable, i.e. a finite imaginary frequency part arises at certain values of the carrier wave length λ= 2 π /k and temperature σ. The four modes, say ω 1, ω 2, ω 3 and ω 4, obtained by solving the dispersion relation (13) are depicted in Figs. 1-4 versus the (reduced) wavenumber k (for 0<k 1) for different values of the temperature σ; we have considered values ranging from 0.1 up to 1, although Landau damping, overseen in the fluid plasma description, should in principle prevail near the upper value. As Figs. 3 and 4 show, the two latter modes ω 3 and ω 4 become imaginary for σ = 0.1, 0.2 and 0.3, that is, we have two growing and decaying modes in the linear approximation. For simplicity, we shall limit the discussion to the level indispensable in the following, leaving a detailed parametric analysis of this multi-fold dispersion law to be reported elsewhere. Note that, naturally, the familiar ion-acoustic dispersion relation ω=±ω IAW (see above) is recovered in the absence of the beam, i.e. upon setting α= 0 in Eq. (13). 1 From Eqs (12) we can express the first-order quantities in terms of φ l as 1 n 1 =A 11 φ 1 1, u 1 1 =B 11 φ 1 1, n 1 b1 =C 11 φ 1 1, 1 u b1 =D 11 φ 1 1, (8) where coefficients A 11, B 11, C 11 and D 11 are given in the Appendix. The second-order (n =2) equations for the 1 st compatibility condition: harmonics (l=1) lead to the following

7 7 λ= ω k k 2 ω k 1+α ω2 k α 2 3σ μ ω 2 k b0 3 u. It is easy to show that λ in fact equals the group velocity λ=v g k =dω/dk, as may readily be checked from the dispersion relation (7). One thus obtains the 2nd-order 1 st -harmonic quantities (9) n 1 2, u , n b1, u b1 in terms of φ and φ 1 / ξ as n φ 1 1 =A 21 ξ +A' 21 φ 2, 2 1 u φ 1 1 =B 21 ξ +B' 21 φ φ 1 n b,1 =C 21 ξ +C' 21 φ 1 2, 2 u b,1 2, 1 1 φ 1 =D 21 ξ +D' 21 φ 2, 2 1 φ φ 1 1 =E 21 ξ +E' 21 φ 2. (10) 1 From the second-order (n =2) reduced equations for l=2, we obtain the second harmonic amplitudes n 2 2, u , n b2, u b2 and φ 2 2, in terms of the quantity φ : 2 n 1 2 A 22 φ , u 2 B 22 φ 1 2 2, n 1 b2 C 22 φ 1 2, 2 u 1 b2 D 22 φ 1 2 2, φ 1 2 E 22 φ 1 2. (11) The coefficients A 22, B 22, C 22, D 22 and E 22 are given in the Appendix. A zeroth-harmonic mode, for l=0, is also present; the corresponding amplitudes cannot be determined completely within the second order, so we have to resort to the 3rd-order equations. Thus, the n=3 and l=0 components of the reduced evolution equations yield: 2 n 1 o =A 20 φ 1 2 2, u 1 bo =B 20 φ 1 2, 2 n bo =C 20 φ 1 1 2, 2 u 1 bo =D 20 φ 1 2 2, φ 1 o =E 20 φ 1 2, (12) where, once more, the coefficients A 20, B 20, C 20, D 20 and E 20 are given in the Appendix. (ii)the nonlinear Schrödinger equation By combining all of the above expressions into the l=1 component of the third-order (n =3) part of the system of reduced evolution equations, we obtain an explicit compatibility condition in the form of a nonlinear Schrödinger equation (NLSE) i φ τ P 2 φ ξ 2 Q φ 2 φ =0, (13)

8 8 which describes the slow evolution of the first-order amplitude of the plasma electric potential 1 perturbation φ φ 1. The dispersion coefficient P reads 1+α ω λ 2 k 3 ω2 k λ 9σ ω ω3 3 σλ 2 k k 3 P= 3 2 k 3 ω2 k 3σ 2 α ω k 2 λ ω k 2 u bo 3 λ ω 2 k 3 μ ω k bo 4 u k λ 2 k. P is in fact related to the dispersion curve as P=d 2 ω/2 dk 2. The nonlinearity coefficient Q, which is due to the carrier wave self-interaction, is given by Q= ω k λ 2 k { 1 ω2 k 2 3σ [ } ω k A 11 B 22 +A 22 B 11 +A 20 B 11 +A 11 B 20 1+α B 22 B 11 +B 20 B 11 3σ A 11 A 22 +A 11 A 20 ] 1+α 1 ω D 22 u k b0 C 11 +D 11 C 22 +D 20 C 11 +D 11 C 20 ωk λ E 2 k 22 +E (15) All parameters were defined in the text above (also see in the Appendix). The perplex form of the coefficients P and Q allows for a parametric investigation in terms of the carrier wavenumber k, as well as the parameters σ, α and u b0, as we shall see in the following Section. For the sake of gaining some analytical insight in the dynamics, one may consider the long wavelength (small wavenumber) behavior of the above quantities. In the limit k << 1, the dispersion relation (13) reduces to ω j c s,j k, where the sound speeds c s,j (for j=1, 2, 3, 4 ) are (14) given by lengthy expressions omitted here; naturally, one recovers c s 1 3 σ = lim k 0 ω IAW k in the vanishing beam limit α 0. Naturally, the group velocity λ given by Eq. (9) tends to the phase velocity ω/ k in this limit, thus confirming the acoustic character of these modes. The dispersion and nonlinear coefficients P and Q, given by relations (14) and (15), respectively vary as P p 0 k and Q q 0 /k versus the wavenumber k in the limit k 0; again, the lengthy expressions for p 0 and q 0 (in fact functions of σ, μ and α ) are omitted here. In consequence,

9 9 the PQ product is finite at long wavelengths (cf. the analysis below), while the coefficient ratio P/Q (which determines the width L of envelope solitons via L ~ P /Q 1/2, as we shall see in the following) varies as ~ k 2. The spatial extension of envelope excitations therefore decreases linearly with k, as k 0. (iii) Envelope soliton solutions of the NLSE The localized solutions of the NLSE (13) describe (arbitrary amplitude) nonlinear excitations, in the form of bright and dark (black/gray) envelope solitons. Exact expressions for these envelope structures can be found by substituting with φ= ρexpiθ into Eq. (13), and then separating real and imaginary parts. The detailed derivation is exposed e.g. in Ref. 30, and has been summarized in Refs The analytical expressions will therefore be omitted here (refer to the expressions and figures in Ref. 32 for a brief summary). Providing brief resume of the properties of envelope solitons is nevertheless imposed, for our purpose here. The bright-type solutions (bright envelope solitons) are obtained for a positive value of the coefficient product PQ, i.e. for PQ > 0. They represent a localized amplitude pulse, which contains (modulates) the fast carrier wave oscillation. This bell-shaped excitation (which vanishes at both infinities) is characterized by an internal slow oscillation ( breathing effect), and bears a maximum amplitude, say φ 0, and a spatial extention (width) L, which are related as Lφ 0 ~ P /Q 1/2 (= constant). This is the identifying signature of these envelope excitations (see in Ref. 32 for a discussion), e.g. in contrast with Korteweg-deVries solitons (which would have φ 0 L 2 =constant, instead). The dark-type solutions (dark envelope solitons) are obtained for a negative value of the coefficient product PQ, i.e. for PQ < 0. They represent a localized amplitude dip, i.e. a hole (a potential void) against an elsewhere constant (finite everywhere) amplitude, which modulates the fast carrier wave oscillation. These excitations may either present a vanishing potential value in the center, i.e. at x=0 (black soliton), or a finite, i.e. non-zero one (gray soliton); in both cases, the asymptotic values at infinity are constant (and finite, i.e. non-zero). Like with bright solitons, the maximum amplitude, say φ 0, and the spatial extention (width) L of dark excitations satisfy Lφ 0 ~ P /Q 1/2 (= constant).

10 10 IV. Modulational stability: numerical investigation and discussion (i) Prerequisites In this section, we present numerical results for the linear stability of amplitude modulated waves. The standard stability analysis consists in linearizing around the monochromatic wave solution of the NLSE (17): ψ= ψ e iq ψ 2 τ +c.c. (c.c. denotes the complex conjugate; note the amplitude dependence or the frequency) by setting ψ= ψ 0 +ε ψ 1, and taking the perturbation ψ 1 to be of the form: ψ 1 = ψ 1,0 e i k ξ ω τ +c. c (one should distinguish the perturbation wave number k and frequency ω from the carrier wave quantities, k and ω ). Substituting into (23), one obtains the dispersion relation ω 2 =p 2 k 2 k 2 2 Q P ψ 0 2. The wave will obviously be stable if the product PQ is negative. However, for positive PQ>0, instability sets in for wave numbers below a critical value k cr = 2Q / P ψ 0, i.e. for wavelength above a wavelength threshold λ cr =2 π / k cr ; defining the instability growth rate γ= Im ω k, we see that it reaches its maximum value for k = k cr / 2, namely γ max = Q ψ 0 2. We draw the conclusion that the instability condition depends only on the sign of the PQ, which may now be studied numerically, relying on the exact expressions derived above. (ii) Numerical analysis As we saw above, the dispersion relation (7) is quartic in the carrier frequency ω, and shows that the inclusion of an electron beam (in addition to a finite ionic temperature) gives rise to four ion-acoustic modes propagating with different phase velocities. In the following, we shall carry out a numerical investigation of the stability of these four acoustic modes. The results of our calculations for the first mode ω 1 k are shown in the ranges of 0 k 1 and 0 σ 1 for α= and v b0 =1.1 in Figs. 5(a)-5(d). Fig 5a shows the dispersion coefficient P 1 versus k and σ ; we remark that P 1 is positive everywhere. So, the stability profile will depend only on the sign of the nonlinear coefficient Q 1 which has been depicted in Fig. 5b; the area in black/white represents the region in the (k, σ ) plane where Q 1 is

11 11 negative/positive, i.e. where wave is stable/unstable. Now we consider the envelope soliton width L 1 which depends on P 1 and Q 1, namely as L 2 1 P 1 /Q 1 (see in the previous Section). Investigating the width of envelope solitons, we find that the width of dark and gray excitations remains constant as σ increases, meanwhile it increases and tends to infinity (when Q 1 becomes zero) as k increases (see Fig. 5c). The behavior of the width of bright excitations is similar to that of black and gray envelope solitons (see Fig 5c-5d). The results of our calculations for the second mode ω 2 k are depicted, in the range of 0 k 1 and 0 σ 1, for α= and v b0 =1.1, in Fig. 6(a)--6(d). Fig 6a shows the dispersion coefficient P 2 versus k and σ from which is seen that P 2 is negative everywhere; so for this mode, the stability and instability depends on the sign of the nonlinearity coefficient Q 2 only. Fig. 6b shows Q 2 versus k and σ ; the area in black/white represents the regions in the (k, σ ) plane which Q 2 is positive/negative, i.e. where the wave is stable/unstable. Notice the small instability region, for large values k and σ, where bright envelope structures may occur. For this mode, the width ( L 2 2 P 2 /Q 2 ) of white/dark and gray excitations behaves in a similar manner to that of the first mode (cf. Figs. 6c-6d and Figs.5c-5d). It may be pointed out that, for some critical values of k and σ which correspond to Q=0, there are no envelope excitations to be derived from Eq (13) (which then becomes linear). In this case, one must consider higher order nonlinearity which has been neglected in the present work. The results of our calculations for the third mode ω 3 k are depicted in Figs. 7a-7c. For this mode, as Figs. 7a-7b show, there are exist only dark and grey envelope solitons, since P 3 and Q 3 have different signs; in fact P 3 >0, Q 3 <0. The area in white which appears in both Figs. 7a- 7b correspond to P 3 and/or Q 3 being complex (i.e. possessing a non-vanishing imaginary part) so that stable wavepacket propagation will be prohibited (damped); as a matter of fact these white areas correspond to complex solutions of the dispersion relation (7). Fig. 7c shows the variations of the width ( L 2 3 P 3 /Q 3 ) of dark (black/gray) envelope solitons (for the third mode) versus k and σ. It is remarked that the width L increases and reaches a maximum value. L decreases as σ increases. Finally, the maximum value of the L increases with σ, for fixed k. Finally, we consider the fourth mode ω 4 k. Figs. 8a, 8b and 8c depict the dispersion and nonlinearity coefficients P 4 and Q 4 and the width ( L 2 4 P 4 /Q 4 ), respectively. In Fig. 8a, the area in full white corresponds to complex solutions of the dispersion relation (7), so that one expects no envelope excitations to occur for this mode; the area in black near σ> 0. 05

12 12 corresponds to P 4 <0 and the area in white corresponds to P 4 >0. There is region nearly 0.3<σ<0.55, in which the determination of the sign of the dispersion coefficient P 4 is quite complicated. The nonlinear coefficient Q 4 has been depicted in Fig 8b; nearly for σ>0.55 and 0.1<σ<0.3, we have modulational stability, i.e. only dark-type envelope excitations. Fig. 4c shows the ratio P 4 / Q 4 (L square) versus k and σ ; we see that the width L increases as k increases for large values σ. V.Conclusions We have investigated the modulational (in)stability profile of ion-acoustic waves, in addition to the occurrence of localized envelope solitons, in a collisionless plasma consisting of warm ions, hot isothermal electrons and a cold electron beam. The effect of the electron beam has been traced via intrinsic plasma parameters like the beam-to-background-electron density ratio α, and the ion-to-background-electron temperature ratio σ. The basic set of fluid equations was reduced to a NLS equation by making use of a reductive perturbation (multiple scales) technique. It was shown that the inclusion of a finite ionic temperature in combination with an electron beam gives rise to four electrostatic modes propagating at different phase velocities. Our main results can be summarized as follows: (i) For the first (modified ion-acoustic) mode, both black-/gray- and bright-type envelope excitations may exist, whose width remains constant (increases) as σ (k, respectively) increases. Modulational instability (favoring bright solitons) occurs in a wide area in (k, σ ) plane, namely for short wavelengths (large k), while wave stability (and bright solitons) is (are) witnessed for small k only; this is true for any value of σ. (ii) For the second (modified ion-acoustic) mode, the qualitative picture is analogous; however, modulational stability occurs in a more extended area in (k, σ ) plane, compared to the first mode. (iii) In the case of the third (novel, beam-plasma related) electrostatic mode, the existence of envelope solitons is excluded at low ion temperature, i.e. for σ<0.1 ; dark envelope structures only occur for σ> 0.1. (iv) Finally, in the case of fourth mode (also a new, beam-plasma mode, in fact), the results are quite analogous: for σ<0.1, no excitations exist, while for higher σ values, both dark and bright-type excitations may arise. It should be noted, for rigor, that high ion temperature values may result to the occurrence of strong Landau damping, due to the electrostatic wave's phase velocity being close to the thermal speed. Such a phenomenon, long known from plasma kinetic theory, is inevitably overseen in a fluid plasma picture like ours, and may somehow admittedly invalidate our results for high σ.

13 13 The present investigation may be significant in understanding the properties of ion-acoustic envelope solitons in space observations as well as plasma laboratory experiments. The effect of an external magnetic field is neglected in this paper, but is currently under investigation. Acknowledgements The work of one of us (I.K.) was partially supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany) through the Sonderforschungsbereich (SFB) Universelles Verhalten Gleichgewichtsferner Plasmen: Heizung, Transport und Strukturbildung.

14 14 References [1] Ya. L. Al pert, Space plasma (Cambridge University Press, New York, 1990). [2] See e.g. in E.Witt and W. Lotko, Phys. Fluids 26, 2176 (1983); also references 1 to 7 therein. [3] Y.Omura, H. Kojima, T. Umeda and H.Matsumoto, Astrophys. Space Sci. 277, 45 (2001). [4] Y. Nejoh, Phys. Fluids B 4, 2830 (1992). [5] Y.Omura, H. Kojima and H.Matsumoto, Geophys. Res. Lett 21, 2943 (1994). [6] Y. Omura, T. Umeda and H. Matsumoto, Simulation of Electron Beam Instabilities and Nonlinear Potential Structures (Springer-Verlag, Berlin, 2003). [7] R. E. Ergun et al., Geophys. Res. Lett. 25, 2061 (1998); G. T. Delory et al., Geophys. Res. Lett. 25, 2069 (1998); R. Pottelette et al., Geophys. Res. Lett. 26, 2629 (1999). [8] J. P. McFadden et al., J. Geophys. Res. 108 (4) 8018 (2003). [9] M. Temerin, Phys. Rev. Lett. 48, 1175 (1982). [10] R. Boström, Phys. Rev. Lett. 61, 82 (1988). [11] H. Matsumoto et al., Geophys. Res. Lett. 21, 2915 (1994); J. R. Franz et al., Geophys. Res. Lett. 25, 1277 (1998); C. A. Cattell et al., Geophys. Res. Lett. 26, 425 (1999). [12] N. Yajima and M. Tanaka, T. Itoh, T. Nakayama, T.Takeda and T.Kamiya, Proc ICPP (Nagoya) 1, 774 (1996). [13] K. Yamagiwa, T. Itoh and T. Nakayama, Journal de Physique IV (Supplément au Journal de Physique III, no. 10) 7, C4-413 (1997). [14] T.Takeda and K. Yamagiwa, J. Plasma Fus. Res. 79 (4), 323 (2003). [15] P. Chatterjee and R. Roychoudhury, Z. Naturforsch. A 51, 1002 (1995). [16] Y. Nejoh and H. Sanuki, Phys. Plasmas 2, 4122 (1995). [17] Y. Nejoh, J. Plasma Phys. 56, 67 (1996); Y. Nejoh, J, Plasma Phys. 57,841 (1997). [18] W. M. Moslem, J. Plasma Phys. 61, 177 (1999). [19] W. M. Moslem, J. Plasma Phys. 63, 139 (2000). [20] A. Esfandyari, S. Khorram and A. Rostami, Phys. Plasmas 8, 4753 (2001). [21] M. Berthomier, R. Pottelette, M. Malingre and Y. Khotyaintsev, Phys. Plasmas 7, 2987 (2000). [22] Quanming Lu and Shui Wang, Proceedings of ISSS-7, Kyoto (2005). [23] Y. Gell and T. Roth, Plasma Phys. 19, 915 (1977). [24] R.G.Greaves and and C.M.Surko, Phys. Rev. Lett 75, 3846 (1995).

15 15 [25] P. Sulem and C. Sulem, Nonlinear Schrödinger Equation (Springer, Berlin, 1999). [26] L. D. Faddeev and L.A. Takhtjan, Hamiltonian methods in the theory of solitons (Springer- Verlag, Berlin, 1980). [27] Kintner, P. M. J. Vago, S. Chesney, R. L. Arnoldy, K. A. Lynch, C. J. Pollock, and T.E. Moore, Phys. Rev. Lett. 68, 2448 (1992). [28] Vago, J. L., P. M. Kintner, S. W. Chesney, R. L. Arnoldy, K. A. Lynch, T. E. Moore, and C. J. Pollock, J. Geophys. Res (1992). [29] D.Y.Wang and G.L. huang, Chin. Phys. Lett, (2001). [30] R. Fedele, H. Schamel, and P.K. Shukla Phys. Scr. T 98, 18 (2002); R. Fedele, H. Schamel, Eur. Phys. J. B 27, 313 (2002). [31] I. Kourakis and P. K. Shukla, J. Phys. A: Math.Gen. 36, (2003); I. Kourakis and P. K. Shukla, Physical Review E 69, (2004). [32] I. Kourakis and P.K. Shukla, Nonlinear Processes in Geophysics, 12, 407 (2005). [33] A. Esfandyari-Kalejahi and H. Asgari, Phys. Plasmas, 12, (2005). [34] N.A. Krall and A. W. Trivelpiece, Principles of plasma physics (NewYork: McGraw-Hill, 1973). [35] T. Stix, Waves in plasmas (New York: American Institute of physics, 1992). [36] T. Dauxois and M. Peyrard, Physics of solitons (Cambridge University Press, Cambridge, 2005). [37] J.K.Xue, W.S. Duan and He Lang, Chinese Physics. 11, 1184 (2002). [38] S.K. EL-Labany, M.S. Abdel Krim, S.A. El-Warraki and W.F. El-Taibany, Chinese Phys. 12, 759 (2003). [39] W. Watanabe, J. Plasma. Phys. 14, 353(1975); H. Bailung and Y. Nakamura, J. Plasma Phys. 50, 231 (1993); Q Z. Luo and N. D. Angelo and R. Merlino, Phys. Plasma. 5, (1998); Y. Nakamura and H. Bailung and P. K. Shukla, Phys. Rev. Lett. 83, 8, 1602 (1999); Y. Nakamura and A.Sarma, Phys. Plasma. 8, 3921 (2001). [40] A. Hasegawa, Phys. Rev. A. 1, 1746 (1970); Idem, Phys. Fluid. 15, (1972). [41] T. Taniuti and N. Yajima J. Math. Phys. 10, 1369 (1969); N. Asano, T. Taniuti and N. Yajima, J. Math. Phys. 10, 2020 (1969).

16 16 Appendix A1. Reduced amplitude evolution equations The evolution equations for the reduced n-th order l-th harmonic amplitudes defined in III.(i) read: n l n 2 τ n ' =1 u l n 2 τ λ n n 1 l ξ l ' = n 1 n il ωn u l 1+α l n 1+α ilku ξ l [ n n ' n n ' ξ l ' ul l 1 n ' +ilku ' l ' nl l ' λ u n 1 l ξ [u 1 1 ' n ' =1 l ' = n 2 n b,l τ n ' =1 n 2 u b,l τ n ' =1 n n ' 1 u n ' l ' ξ λ n n 1 b,l ξ l ' = ξ u b,l l ' λ u n 1 b,l ξ u b,l l ' l ' = 2 φ n 2 ξ n ' =1 n n ' ] =0, il ωu n φ n 1 l n +ilk φ l ξ l 3σ +il ' ku n n ' n ' l l ' ul ' 3σ 1+α n 2 l l ' n il ωn b,l α u n 1 b,l +ilα ku ξ b,l n ' n n ' n ' +ilkub,l l nb,l ' =0, n n ' 1 nbl '' n n ' 1 n n 1 l 1+α ξ n n n ' 1 n ' l ' ξ n 1 n n +u b,l b0 ξ n +ilk n l ' ] +il ' kn n n ' n l l ' nl ' =0, n +ilk u b0 n b,l n 1 il ωu n u b,l +u b,l b0 +ilu ξ b0 u n b,l 1 n 1 φ l 1 μ ξ μ ilk φ n l n ' +il ' ku n n ' n ' b,l l ' ub,l ' =0, ξ n b,l '' 2lik φ n 1 l ξ l ' = φ n n ' n ' l l ' φl ' 1 6 n ' =1 l 2 k 2 n φ l φ n n l n n b,l +n l n '' =1 l ' = l ' = φ n n ' n '' φl ' l l ' l '' n ' n φ '' l '' =0. (A1.1) (A1.2) (A1.3) (A1.4) (A1.5)

17 17 A2. Coefficient definitions in the perturbative solution The coefficients appearing in the solutions aobtained for the reduced n-th order l-th harmonic amplitudes defined in III.(i), namely in Eqs. (8) and (9-12), are provided in the following. A2.1 n=1, l=1 coefficients: A 11 = 1+α ω 2 k 2 3σ, B 11 = ω ω2 k 2 3σ k, C 11 = α μ ω k b0 2 u, D 11 = 1 μ ω k b0 u. A2.2 n=2, l=2 coefficients: 3 1+α ω2 k +σ 2 A22= 2 ω2 3 σ 3 2 k +A11 E 22 B 22= ω ω2 k 2 9 σ 2 k ω2 3 σ 3 2 k +B 11 E 22 3α 1 C 22= 2 μ 2 ω +C 11 E 22 D 22= k νb0 4 2 μ 2 ω +D 11 E 22 k νb0 3 1+α ω2 k +σ 2 α E 22= 2 k 2 ω2 3σ 3 2 k 2 μ 2 ω 1 2 k k νb0 3 4 A2.3 n=2, l=0 coefficients: A20= 1+α ω2 k 2 λω 2 k 3σ 3σ λ 2 ω2 3σ 2 2 k E 20 3σ λ 2 6σω +λ k B 20= ω2 k 3σ 2 3σ λ 2 ω2 3σ 2 2 k λe 20 3σ λ 2

18 18 α ω 3νb0+ 2 λ k C 20= μ 2 vb 0 λ 2 ω k vb 0 3 αe 20 D 20= μ vb 0 λ 2 1 μ vb 0 λ 1 μ ω +E k 20 vb E α 20= α ω 3 vb 0 2 λ 1+α ω2 k k 2 λω 2 k 3σ 1+α μ vb 0 λ 2 3σ λ 1 μ 2 vb 0 λ 2 ω 2 k vb 0 3 3σ λ 2 ω2 3σ k A2.4 n=2, l=1 coefficients: A 21 = C 21 = 2 1+α ω k ω k λ 2, A' 21 = ik ω2 k 1+α ω2 3σ k 3σ, B 21 = α ω k λ ikμ ω k u b0 3, 2 C' 21 = α μ ω k u b0, D 21 = ω k λ ω2 k +σ 2 2, B' 21 = ik ω2 k 3σ 2 ω k λ ik μ ω k u b0 2, ω k ω2 k 3σ 2 1 D' 21 = μ ω k b0 u.,

19 19 Figure captions Figure1 Dispersion relation for the 1 st ES mode: the mode frequency ω 1 is depicted versus the (reduced) wavenumber k, for different values of σ ; here, we have considered α= and v b0 Figure 2 Dispersion relation for the 2 nd ES mode: the mode frequency ω 2 is depicted versus the (reduced) wavenumber k, for different values of σ ; here, α= and v b0 Figure 3 Dispersion relation for the 3rd ES mode: the mode frequency ω 3 is depicted versus the (reduced) wavenumber k, for different values of σ ; here, α= and v b0 Figure 4 Dispersion relation for the 4th ES mode: the mode frequency ω 4 is depicted versus the (reduced) wavenumber k, for different values of σ, for α= and v b0 Figure 5 Fig. 5a A three dimensional plot of the dispersion coefficient P 1 versus σ and the (normalized) wavenumber k/ k D, in the ranges 0<k/ k D <1 and 0 σ 1. Here, α= and v b0 See that P 2 >0 everywhere.

20 20 Fig. 5b The Q 1 =0 contour is depicted against the normalized wave-number k/ k D, for 0 < k/ k D < 1, and the temperature ratio σ, 0 σ 1 : black/white represents the region where Q 1 is negative/positive, i.e. the region of modulational stability/instability, which may support dark/bright-type solitary excitations. Here α=0.002 and v b0 Fig. 5c A three dimensional (3D) plot of the width of dark solitons P 1 / Q 1 < 0 versus σ and k/ k D, in the ranges 0 < k/ k D < 1 and σ =1/5, 2/5, 3/5, 4/5 and 1; here α=0.002 and v b0 Fig. 5d A 3D plot of the width of bright-type solitons, where P 1 / Q 1 > 0 versus σ and k/ k D, in the ranges 0 < k/ k D < 1 and σ =1/5, 2/5, 3/5, 4/5 and 1; here α= and v b0 Figure 6 Fig. 6a A 3D plot of the dispersion coefficient P 2 versus σ and the (normalized) wave-number k/ k D, in the ranges 0<k/ k D <1 and 0 σ 1. Here, α=0.002 and v b0 See that P 2 <0 everywhere. Fig. 6b The Q 2 =0 contour is depicted against the normalized wave-number k/ k D, for 0 < k/ k D < 1, and the temperature ratio σ, 0 σ 1 : black/white represents the region where Q 1 is positive/negative, i.e. the region of modulational stability/instability, which may support dark/bright-type solitary excitations. Here α=0.002 and v b0 Fig. 6c A three dimensional (3D) plot of the width of dark solitons P 2 / Q 2 < 0 versus σ and k/ k D, in the ranges 0 < k/ k D < 1 and σ =1/5, 2/5, 3/5, 4/5 and 1; here α=0.002 and v b0

21 21 Fig. 6d A 3D plot of the width of bright-type solitons, where P 2 / Q 2 > 0 versus σ and k/ k D, in the ranges 0 < k/ k D < 1 and σ =1/5, 2/5, 3/5, 4/5 and 1; here α= and v b0 Figure 7 Fig. 7a The dispersion coefficient P 3 =0 contour is depicted against the normalized wave-number k/ k D, for 0 < k/ k D < 1, and the temperature ratio σ, for 0 σ 1 ; black represents the region where P 3 is positive (real), while white represents the region where P 3 is complex. Here, α= and v b0 Fig. 7b The nonlinear coefficient Q 3 =0 contour is depicted against the normalized wave-number k/ k D, for 0 < k/ k D < 1, and the temperature ratio σ, for 0 σ 1 ; black represents the region where Q 3 is negative (real), where dark-type envelope excitations may occur; white represents the region where Q 3 is complex. Here, α= and v b0 Fig. 7c A 3D plot of the coefficient ratio P 3 / Q 3 (<0, here) versus the temperature ratio σ and the normalized wave-number k/ k D, in the range 0 < k/ k D < 1, and for σ =1/5, 2/5, 3/5, 4/5 and 1. Here, α= and v b0 Figure 8 Fig. 8a The dispersion coefficient P 4 =0 contour is depicted against the normalized wave-number k/ k D, for 0< k/ k D <1, and the temperature ratio σ, for 0 σ 1 ; black represents the region where P 4 is negative (real), while white represents the region where P 4 is complex. Here, α=0.002 and v b0 =1.1

22 22 Fig. 8b The nonlinear coefficient Q 4 =0 contour is depicted against the normalized wave-number k/ k D, for 0 < k/ k D < 1, and the temperature ratio σ, for 0 σ 1 ; black represents the region where Q 3 is negative (real), where bright-type envelope excitations may occur; white represents the region where Q 3 is complex. Here, α= and v b0 Fig. 8c A 3D plot of the coefficient ratio P 4 / Q 4 (<0, here) versus the temperature ratio σ and the normalized wave-number k/ k D, in the range 0 < k/ k D < 1, and for σ =14/20, 15/20, 16/20, 17/20, 18/20, 19/20 and 1. Here, α= and v b0

23 23 Figure 1 Figure 2

24 24 Figure 3 Figure 4

25 25 Figure 5 (a) (b) (c) (c)

26 26 Figure 6 (a) (a) (b) (c) (d)

27 27 Figure 7 (a) (b) (c)

28 28 Figure 8 (a) (b) (c)