Blow-up solitons at the nonlinear stage of the two-stream instability in quantum plasmas
aa r X i v : . [ n li n . PS ] J u l epl draft Blow-up solitons at the nonlinear stage of the two-stream insta-bility in quantum plasmas
V. M. Lashkin Institute for Nuclear Research - Pr. Nauki 47, Kyiv 03680, Ukraine
PACS – Solitons
PACS – Quantum transport
PACS – Microinstabilities (ion-acoustic, two-stream, loss-cone, beam-plasma, drift, ion-or electron-cyclotron, etc.)
Abstract –The nonlinear evolution of the quantum two-stream instability in a plasma withcounter-streaming electron beams is studied. It is shown that in the long-wave limit the non-linear stage of the instability can be described by the elliptic nonlinear string equation. Wepresent two types of the nonlinear solutions. The first one is an unstable nonlinear mode that iscontinuously related with the growing linear solution and the second one is a pulsating soliton.We show that both of these solutions blow up in a finite time.
Introduction. –
The two-stream instability is one ofthe most known instability in plasma physics [1]. Thisinstability may appear when the plasma particles streamwith different velocities. In particular, this may be theclassical instability of an electron beam in a plasma [2, 3]and the Buneman instability [4] which are widespread inboth laboratory and space plasmas. In the context ofquantum plasmas, the problem of the two-stream instabil-ity was first studied by Haas et al. [5]. Quantum effects inplasmas are important in the limit of low-plasma tempera-ture and high-particle number density [6,7]. Such plasmasare ubiquitous in microelectronic devices, in dense astro-physical plasma, in microplasmas, and in laser plasmas(for example, see the reviews [8,9] and references therein).There are two well-known models for describing the quan-tum effects in a plasma. The Wigner and Hartree mod-els are based upon the Wigner-Poisson and Schr¨odinger-Poisson systems which, respectively, correspond to thestatistical and hydrodynamic description of the plasmaparticles. Authors [5] have introduced a quantum multi-stream model by using the nonlinear Schr¨odinger-Poissonsystem. In particular, they derived the dispersion rela-tion for the quantum two-stream instability and showedthe existence of new, purely quantum, unstable brancheswhich do not appear in classical plasmas. Later on, thequantum hydrodynamic description of the quantum two-stream instability [5, 10] was extended to the kinetic case[11] which was based on the quantum mechanical Wignerformalism. Since then, the quantum two-stream insta- bility has attracted considerable attention and been dis-cussed by many authors for different plasma regimes suchas dusty quantum plasma [12], electron-positron-ion quan-tum plasma [13], dense electron-ion plasma [14], relativis-tic plasma [15], and electron-hole quantum semiconductorplasma [16]. The effects of the exchange interaction [17]and the electron spin [18] for the quantum two-stream in-stability have been also investigated. Necessary validityconditions for the cold quantum two-stream model werediscussed in ref. [19]. Recently, it was also shown [20] thatthe hydrodynamic model gives a decent approximation ofthe exchange effects for sufficiently long wavelengths.A nonlinear stage of the quantum two-stream instabil-ity was studied back in the original work [5] by numer-ical simulation, where, in particular, spatially periodic,stationary states which survive over long times were pre-dicted. The aim of this Letter is to derive a nonlinearevolutional equation describing the nonlinear stage of thequantum two-stream instability with cold electrons in thelong-wavelength limit and to present corresponding exactanalytical solutions. To our knowledge, this equation inthe form of the elliptic nonlinear string (ENS) equation inthis context was not previously obtained even for classicalplasmas. In the classical plasma, a nonlinear stage of thelong-wavelength Buneman instability was studied in themodel of the Boussinesq equation, where the dispersioncorresponds to warm electrons [21]. The ENS equationdiffers from the classical Boussinesq equation in the signbefore a fourth derivative term. Note that when consid-p-1. M. Lashkin 1ering the nonlinear stage of the quantum two-stream in-stability in electron-positron-ion plasmas the Korteweg-deVries (KdV) equation was obtained by employing the re-ductive perturbation technique [13]. The ENS equationas well as the KdV equation is completely integrable andadmits N -soliton solution [22]. A significant difference,however, is that the dispersion relation of the linearizedKdV equation does not corresponds to any instability atall, while the dispersion relation of the ENS equation com-pletely coincides with the one of the two-stream instabilityin the long-wavelength limit. Model equation. –
The longitudinal dielectric re-sponse function of a cold plasma with the ”quantum re-coil” term for two counter propagating electron beamswith equal equilibrium number densities and velocities hasthe form [5, 7] ε ( ω, k ) = 1 − ω p / ω − kv ) − ~ k / (4 m ) − ω p / ω + kv ) − ~ k / (4 m ) , (1)where ω and k are the frequency and the wave numberrespectively, ω p = p πe n /m is the electron plasma fre-quency, n is the equilibrium plasma density, e is the el-ementary charge, m is the electron mass, ~ is the Planckconstant divided by 2 π , and v is the stream velocity.Charge neutrality is provided by the motionless back-ground ions. The solution of eq. (1) for ω has twobranches [5] ω ± = ω p k v + ~ k m ± ω p s k v ω p + 4 ~ k v m ω p . (2)The solution with the sign ”+” is always positive and givesstable oscillations. The other solution can be negative( ω <
0) and the instability occurs provided that [5][ H K − H K − K + 4] < , (3)where the rescaled variables are H = ~ ω p / ( mv ) , K = kv /ω p . (4)In this Letter, we restrict to the case of the long-wavelimit when K ≪ H K ≪
1. Note that, generallyspeaking, a strongly quantum regime H ≫ − ” takes the form ω = − k v + 4 v ω p (cid:18) H (cid:19) k (5)and predicts the instability with the growth rate γ = kv q − H / k v /ω p . (6)We address the nonlinear stage of this instability. In the following we use the notation X q = q + q · · · → Z · · · δ ( q − q − q ) dq (2 π ) dq (2 π ) , (7)where δ ( x ) is the Dirac delta function and q = ( ω, k ).In the one-dimensional space the Wigner kinetic equa-tion [23–25] for the quantum electron distribution function(Wigner function) F ( x, v, t ) can be written as ∂F∂t + v ∂F∂x = − iem π ~ Z Z dλdv ′ exp h i m ~ ( v − v ′ ) λ i × (cid:20) ϕ (cid:18) x + λ , t (cid:19) − ϕ (cid:18) x − λ , t (cid:19)(cid:21) F ( x, v ′ , t ) , (8)where ϕ is the electrostatic potential. In the momentumspace eq. (8) can be written as( ω − kv ) f q ( v ) = em π ~ Z Z dλdv ′ exp h i m ~ ( v − v ′ ) λ i × h(cid:16) e ikλ/ − e − ikλ/ (cid:17) ϕ q f (0) ( v ′ )+ X q = q + q (cid:16) e ik λ/ − e − ik λ/ (cid:17) ϕ q f q ( v ′ ) (9)where f q ( v ) is the deviation of the electron distributionfunction of each stream from the equilibrium one f (0) ( v ),and ϕ is the electrostatic potential. The distribution func-tion f (0) ( v ) is normalized to the equilibrium plasma den-sity of each stream, R f (0) ( v ) dv = n /
2. Integrating over λ in eq. (9) yieldsexp (cid:20) i m ~ ( v − v ′ ) ± ik (cid:21) λ → πδ (cid:20) m ~ ( v − v ′ ) ± k (cid:21) , (10)and then integrating over v ′ yields( ω − kv ) f q ( v ) = e ~ ϕ q (cid:20) f (0) (cid:18) v + ~ k m (cid:19) − f (0) (cid:18) v − ~ k m (cid:19)(cid:21) + e ~ X q = q + q ϕ q (cid:20) f q (cid:18) v + ~ k m (cid:19) − f q (cid:18) v − ~ k m (cid:19)(cid:21) . (11)We present the function f q ( v ) as a series in powers of thefield strength (i. e. f ( n ) q ∼ ϕ n ) f q ( v ) = ∞ X n =1 f ( n ) q ( v ) . (12)In the linear approximation from eqs. (11) and (12) wehave f (1) q = eϕ q ~ ( ω − kv ) (cid:20) f (0) (cid:18) v + ~ k m (cid:19) − f (0) (cid:18) v − ~ k m (cid:19)(cid:21) , (13)and then one can write the recurrence relation f ( n ) q = e ~ ( ω − kv ) X q = q + q ϕ q (cid:20) f ( n − q (cid:18) v + ~ k m (cid:19) − f ( n − q (cid:18) v − ~ k m (cid:19)(cid:21) . (14)p-2low-up solitons at the nonlinear stage of the two-stream instability in quantum plasmasFor the nonlinear terms ( n >
2) we use the quasiclassicalapproximation and in the limit ~ k/ (2 m ) ≪ v in eq. (14)one can expand f ( n − q (cid:18) v ± ~ k m (cid:19) ≈ f ( n − q ( v ) ± ∂f ( n − q ∂v ~ k m (15)whence we get f ( n ) q = em ( ω − kv ) X q = q + q k ϕ q ∂f ( n − q ∂v . (16)Retaining terms in eq. (12) up to second order in the wavefields and substituting f q into the Poisson equation k ϕ q = − πe X α Z f q ( v ) dv, (17)where P stands for summation over the different electronstreams labeled by α = + , − , and the ion contribution isomitted, we get ε q ϕ q = X q = q + q V q ,q ϕ q ϕ q , (18)where ε q = 1 + 4 πe ~ k X α Z (cid:2) f (0) (cid:0) v + ~ k m (cid:1) − f (0) (cid:0) v − ~ k m (cid:1)(cid:3) ( ω − kv ) dv (19)is the linear electron dielectric response function, and theinteraction matrix element V q ,q is determined by V q ,q = e m ω p n k X α Z k [( ω + ω ) − ( k + k ) v ] × ∂∂v k ( ω − k v ) ∂f (0) α ∂v dv + ( ω , k ⇄ ω , k ) . (20)Note that the expression (20) for the interaction matrixelement V q ,q is written in a symmetrized form. Singular-ities in the denominators in eqs. (19) and (20) are avoided,as usual, using Landau’s rule by replacing ω → ω + i ω − kv ) − = P ( ω − kv ) − − iπδ ( ω − kv ) , (21)[ ω + ω − ( k + k ) v ] − = P [ ω + ω − ( k + k ) v ] − − iπδ [ ω + ω − ( k + k ) v ] , (22)where P is the principal value of the integrals. Imaginaryparts in eqs. (21) and (22) account for the linear and non-linear Landau damping respectively. In this paper we donot take into account the damping and only the princi-pal value of the integrals is understood. After two partialintegrations in eq. (20) one can write V q ,q = e m ω p n k X α Z (cid:20) k k k ( ω − kv ) ( ω − k v )+ kk k ( ω − kv ) ( ω − k v ) (cid:21) f (0) α dv + ( ω , k ⇄ ω , k ) , (23) where ω = ω + ω and k = k + k . Expanding ε ( ω, k )given by eq. (1) near the ω k determined by eq. (5) yields ε ( ω, k ) = ε ( ω k , k ) + ε ′ ( ω k )( ω − ω k ) (24)where ε ′ ( ω k ) ≡ ∂ε ( ω ) /∂ω | ω = ω k and in the leading orderone can get ε ′ ( ω k ) = − ω p k v . (25)After substituting eq. (24) into eq. (18) we have( ω − ω k ) ϕ q = 1 ε ′ ( ω k ) X q = q + q V q ,q ϕ q ϕ q . (26)Considering the monoenergetic electron streams, v ≫ v T e , where v T e is the thermal electron velocity, one canwrite for the full electron distribution function f (0) f (0) = f (0) − + f (0)+ = n δ ( v − v ) + δ ( v + v )] . (27)Substituting eq. (27) into eq. (19) and then after suitablechanges of variables one can obtain the dielectric function(1). Substituting eq. (27) into eq. (23) we obtain V k ,k = e m ω p k (cid:20) k k k (cid:18) − Ω − , + 1Ω Ω + , (cid:19) + kk k − Ω − , + 1Ω Ω , ! + ( ω , k ⇄ ω , k ) , (28)where Ω ± = ω ± kv , Ω ± , = ω ± k v . The wave dis-persion in eq. (5) has an acoustic type and in the leadingterm satisfies the three-wave resonance condition ω k = ω k + ω k , k = k + k . (29)Taking into account eq. (5) and eq. (29) when calculating(28), and then substituting eq. (25) and eq. (28) into (26)we finally obtain (cid:20) ω + k v − v ω p (cid:18) H (cid:19) k (cid:21) ϕ q = − em k X q = q + q ϕ q ϕ q . (30)In the physical space this equation takes the form ∂ ϕ∂t + v ∂ ϕ∂x + 4 v ω p (cid:18) H (cid:19) ∂ ϕ∂x + 3 em ∂ ϕ ∂x = 0 . (31)In the linear approximation, taking ϕ ∼ exp( − ikx + iωt ),the equation (31) yields the dispersion relation (5). Afterrescaling t → ω p t, x → ω p v x, ϕ → eϕ mv (1 + H / , (32)p-3. M. Lashkin 1equation (31) can be written in the following dimensionlessform ϕ tt + ϕ xx + 4 (cid:18) H (cid:19) ( ϕ xx + 6 ϕ ) xx = 0 . (33)The linear part of this equation corresponds to the ellip-tic operator. The sign before the nonlinear term is notessential and can be changed by replacing ϕ → − ϕ . Theequation (33) with negative signs before ϕ xx and ϕ xxxx isthe classical Boussinesq equation and its N -soliton solu-tions were obtained by Hirota [26] using his bilinearizationmethod. The linear dispersion relation of the Boussinesqequation predicts the instability for short waves unlike theequation of a nonlinear string (33) describing the long-wave instability. Yadjima pointed out [27] that in the Hi-rota’s solutions also implicitly present the solutions whichare continuously related to the unstable solution of thecorresponding linear equation. The solution in [27] de-scribes the nonlinear stage of the linear instability. Underthis, the over-stabilization takes place the unstable modeneither grows unlimitedly nor saturates at some level buttakes the maximum and thereafter damps. As we see be-low, the situation is essentially different for eq. (33). Blow-up nonlinear solutions. –
The equation (33)with negative sign only before ϕ xx is known as the equa-tion of nonlinear string. Although this equation is linearlystable, Kalantarov and Ladyzhenskaya [28] were appar-ently the first to show that real solutions with initial datasatisfying the inequality H <
0, where H is the hamilto-nian, blow up in a finite time. For the same equation thepossibility of collapsing solitons with H > ϕ = k {√ P cosh[ γt + ln( a √ P / k )] cos kx − }{√ P cosh[ γt + ln( a √ P / k )] − cos kx } (34)where γ = k p − H / k , P = 1 − H / k − H / k . (35)Here, a and k are arbitrary parameters satisfying the con-ditions a > H / k < P >
0. Byexpanding eq. (34) in a one can obtain ϕ = a exp( γt ) cos kx + a k exp(2 γt ) cos 2 kx + 3 a k exp(3 γt ) (cid:20) cos 3 kx − k γ cos kx (cid:21) + . . . , (36) which is in agreement to every order in a with the expres-sion obtained from the perturbation theory. At the sametime, neglecting the nonlinear term in eq. (33) yields thegrowing solution ϕ = a exp( γt ) cos kx (37)with the growth rate γ given by eq. (6) and corresponds tothe linear dispersion relation (5). Thus, the solution (34)is continuously related to the linear solution (37). Since P < t cr = 1 γ ln 2 k a (1 + √ − P ) . (38)This is in contrast to the behavior predicted in ref. [27] forthe Boussinesq equation, when the unstable solution ini-tially grows with time then reaches the maximum at sometime and thereafter damps so that the over-saturating ofthe instability takes place.Another exact solution of eq. (33) is the pulsating(breather) soliton. One can obtain this solution justputting γ = i Ω, where Ω is real, and a = 2 k / √ P ineq. (34), so that there is only one arbitrary parameter q in the solution ϕ = q (1 + √ Q cos Ω t cosh qx )( √ Q cos Ω t + cosh qx ) , (39)whereΩ = q p H / q , Q = 1 + 16(1 + H / q H / q . (40)It is seen that Q > t cr = 1Ω arccos (cid:18) − √ Q (cid:19) . (41)It is seen that in the strongly quantum regime H ≫ H →
0, we readily obtainthe case of classical plasma, which for the two cold coun-tering electron streams was also not discussed earlier inthe literature in this context.A singularity in the solutions indicates that the model(31) is no longer valid and the full model [5] should beused. In reality, the singularity can never be reached andthe blow-up is prevented by including extra effects such asLandau damping, higher-order dispersion and nonlinear-ities. In our case of the monoenergetic electron streams,the stabilization can occur long before thermalization ofthe beams or reaching the characteristic scale length ofthe order of the Debye radius due to the electron trappingin wave electrostatic potential well when eϕ ∼ mv [30]which is in agreement with [5].p-4low-up solitons at the nonlinear stage of the two-stream instability in quantum plasmas Conclusions. –
We have derived the nonlinear equa-tion describing the development of the quantum two-stream instability in cold plasmas with counter streamingelectrons. This equation has a form of the elliptic nonlin-ear string equation. We have presented two type of theexact solutions of this equation. The first one is an un-stable nonlinear mode that continuously related with thegrowing linear solution and the second one is a pulsatingsoliton. We have shown that both of these solutions blowup in a finite time.
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