Dressed ion acoustic solitary waves in quantum plasmas with two polarity ions and relativistic electron beams

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Dressed ion acoustic solitary waves in quantum plasmas with two polarity ions and relativistic electron beams Yunliang Wang a,b,∗ , Yushan Dong a , B. Eliasson b a

Department of Physics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China International Centre for Advanced Studies in Physical Sciences and Institute for Theoretical Physics, Faculty of Physics and Astronomy, Ruhr University Bochum, D-44780 Bochum, Germany b

a r t i c l e

i n f o

Article history: Received 25 June 2013 Received in revised form 6 August 2013 Accepted 14 August 2013 Available online xxxx Communicated by F. Porcelli Keywords: Quantum plasma Quantum ion acoustic waves Relativistic electron beams

a b s t r a c t A theory for dressed quantum ion acoustic waves (QIAWs), which includes higher-order corrections when QIAWs are investigated by the reductive perturbation method, is presented for unmagnetized plasmas containing positive and negative ions and weakly relativistic electron beams. The properties of the QIAWs are investigated using a quantum hydrodynamic model, from which a Korteweg–de Vries equation is derived using the reductive perturbation method. An equation including higher-order dispersion and nonlinearity corrections is also derived, and the physical parameter space is discussed for the importance of these corrections. © 2013 Elsevier B.V. All rights reserved.

1. Introduction

Quantum effects in plasmas have important applications to the next-generation intense laser–solid density plasma interaction [1], quantum X-ray free electron lasers [2,3], semiconductors and micromechanical devices, and compact astrophysical object [4] such as white dwarf stars and neutron stars [5]. Due to the Pauli exclusion principle for fermions, quantum effects become important when the de Broglie wavelength of the electrons is comparable to the inter-electron distance. Due to the electrons degeneracy at equilibrium, they will obey the Fermi–Dirac distribution instead of the Boltzmann distribution. Similar to classical plasmas, where fluid equations for electrostatic waves can be derived via moments of the Vlasov–Poisson system, quantum hydrodynamic (QHD) equations including the Bohm potential due to the quantum tunneling effect can be obtained via moments of the Wigner–Poisson system [6]. Linear and nonlinear collective excitations in quantum plasmas have been investigated including the effects of quantum tunneling, quantum statistic, as well as electron spin on the electrostatic acoustic and electromagnetic waves [7,8]. Among them, one of the most important nonlinear waves is the quantum ion acoustic wave (QIAW) which has applications to carbon nanotubes [9,10] and other lab-

*

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oratory and astrophysical settings, and which can be investigated using a QHD model for the inertial ions and inertialess electrons [11]. For QIAWs with small but finite amplitudes, a Korteweg–de Vries (KdV) equation can be derived and used to investigate quantum effects on ion-acoustic solitary waves [11,12]. The nonlinear wave modes are characterized by a quantum parameter H , which is proportional to the ratio between the plasmon energy h¯ ω pe and the Fermi energy k B T F e with h¯ being the Plank constant divided by 2π , k B Boltzmann’s constant, T F e the electron Fermi temperature, and ω pe the electron plasmas frequency. For cylindrical and spherical geometry, the QIAW can be described by a modified KdV [13] or modified nonlinear Schrödinger [14] equation, while for QIAWs with arbitrary large amplitudes, pseudopotential techniques can instead be used [15]. The modulational instability of two-dimensional QIAW packets is strongly influenced by the quantum parameter at small scales [16]. For double layer structures of QIAWs, the quantum diffraction parameter reduces the steepness [17]. Quantum ion-acoustic shock waves can be formed in the presence of ion kinematic viscosity, but the shock can break up into solitary waves when the effects of viscosity are small [18]. Shock waves in cylindrical or spherical geometries have also been investigated [19]. Monopolar, dipolar, and vortex street-type QIAWs can form in magnetized quantum plasmas, where the quantum diffraction effects modifies the length scales of the vortices [20]. Quantum plasmas with positive and negative ions have also attracted much attention due to great potential applications of negative ions in microelectronic or photoelectronic industries [21]. Negative ions can also be produced by electron attachment to

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neutral particles or may be injected from external sources. The linear and nonlinear propagation of QIAWs have been investigated in the quantum pair-ion plasmas by considering the inertial ions and inertialess electrons [22]. Recently, the effects of negative ions on shock structures were also considered in a degenerate dissipative plasma, where the strength of the shock wave increases with the decrease of the negative ion density [23]. The phase shifts in the head-on collision of quantum dust ion acoustic solitons are also affected by the effects of quantum parameter H and the ratio of charge density of positive ions to that of negative ions in quantum pair-ion plasmas [24]. On the other hand, the physics of beam plasmas system have received a great deal of interest in understanding the basic properties of magnetospheric and solar physics. Nonlinear ion acoustic waves (IAWs) can be excited when an electron beam is injected into a plasma. In a classical plasmas with hot isothermal beam and electrons and warm ions, there are four ion acoustic waves modes and the corresponding rarefactive solitons with small amplitude waves [25]. While for IAWs with arbitrary amplitude, a pseudopotential method can be used to investigate the effect of electron beams on the IAWs, which shows that the existence of IAWs sensitively depend on density of electron beams that can reduce the propagation speed of the IAWs [26]. The modulational instability of the four distinct IAW modes has different behavior from each other, where the stability criteria depend on the velocity and density of the beam [27]. More recently, IAWs in an electron beam superthermal plasmas were investigated in the fully nonlinear regime, which showed that solitary waves with both negative and positive polarities can coexist, and that the speeds of the solitary waves depended on the temperature of the ions and superthermal electrons [28]. Recently linear instability for two stream quantum plasma was studied by multistream model derived from Klein– Gordon–Maxwell system [29]. The relativistic steam instability are important for the electron heating in intense laser plasma system [30], in rotating thermal viscous objects [31], and in astrophysical relativistic shocks [32]. However, very little work has been done on the effect of an electron beam on nonlinear QIAWs in a quantum plasma. We here investigate the QIAWs in quantum plasmas with two polarity ions, and relativistic electron beams. Higher-order nonlinearities modifies the solitary wave amplitude and can change its shape [33]. Ion acoustic solitons with higher-order corrections, called dressed solitons, was considered by including higher-order terms in the reductive perturbation method, and by using multiple space–time variables to derive the higher-order terms [34]. Dressed solitons obtained numerically contained high-frequency Langmuir field envelopes together with potential depressions that became oscillatory away from the interaction region [35]. Recently dressed soliton in quantum dusty pair-ion plasmas [36] and electron–positron– ion plasmas [37] have also been studied using a higher-order inhomogeneous differential equation, but without considering the effects of electron beams. Higher-order corrections to nonlinear waves were investigated by considering the effects of electron beams on the properties of compressive or rarefactive solitons [38] in classic plasmas. For quantum ion acoustic solitary waves with very small amplitude, the KdV equation with lowest-order nonlinearity and dispersion is good enough to describe the solitary waves. As the amplitude increases, the amplitude and width of solitons will deviate strongly from the prediction of the KdV equation [39]. Accordingly the higher-order nonlinear and dispersion effects must also be considered. In this Letter, we will investigate the properties of dressed QIAWs in the quantum plasma and to consider the combined effects of quantum diffraction, relativistic electron beams, and negative ions on the propagation of nonlinear QIAWs. By using numerical analysis we will discuss the physical condition to include the high-order corrections.

2. Basic equations Let us consider a quantum plasma consisting of two polarity ions, and electron beams. The charge neutrality at equilibrium reads n+0 = ne0 + n−0 , where n+0 , n−0 , and ne0 , are the equilibrium number densities of the positive and negative ions, and electron beam, respectively. Collisions have been neglected between electrons and the two types of ions. We will assume that the particles obeys the one-dimensional zero temperature Fermi gas, and obey the pressure law p j = (m j v 2F j /3n2j0 )n3j [6], with j = e , +, − standing for electrons, positive ions, and  negative ions, respectively, and where m j is the mass and v F j = 2k B T F j /m j is the Fermi speed. The set of equations describing the dynamics of positive ions and negative ions are given as

∂ n+ ∂ n+ v + + = 0, ∂t ∂x ∂φ ∂ v+ ∂ v+ ∂ n+ + v+ =− − σ+n+ , ∂t ∂x ∂x ∂x ∂ n− ∂ n− v − + = 0, ∂t ∂x ∂ v− ∂ v− ∂φ ∂ n− + v− = μ− − σ−n− , ∂t ∂x ∂x ∂x

(1) (2) (3) (4)

where n+ and n− are the number densities of the positive and negative ions, normalized by their equilibrium number densities n+0 and n−0 , respectively. Also, v + and v − are the fluid velocities of the positive and negative ions, normalized by the quantum  ion acoustic speed c s+ = 2k B T F e /m+ , where k B is Boltzmann’s constant, T F e is the electron Fermi temperature, σ+ = T F + / T F e is the ratio of the Fermi temperatures of the positive ions and electrons, σ− = μ− T F − / T F e is the combined effects of the ratio of positive ion mass to the negative ion mass μ− = m+ /m− and the ratio of Fermi temperature of negative ions to that of electrons. On the ion time scale, the electron beams with a streaming velocity much larger than electron Fermi speed can be described by the fluid equations. We also have the following two equations for the beam electrons as

∂ ne v e ∂ ne + = 0, ∂t ∂x   ∂ ∂φ ∂ ne ∂ ( v e γe ) = σe + ve − σe ne ∂t ∂x ∂x ∂x  √  H e2 cn4 ∂ 1 ∂ 2 ne , + √ 2γe ∂ x ne ∂ x2

(5)

(6)

where γe = (1 − v e2 /cn2 )−1/2 is the relativistic gamma factor for the electron beams. Here we have for convenience normalized the electron beam velocity v e and speed of light in vacuum cn by the quantum ion acoustic speed c s+ . In the weakly relativistic limit, the gamma factor can be written γ = 1 − v e2 /2cn2 . Here σe = m+ /me is the mass-ratio between positive ions and electrons. The quantum parameter is H e = h¯ ω p + /me c 2 for electrons. The electrostatic potential φ can be determined by Poisson’s equation

∂ 2φ = μn− + (1 − μ)ne − n+ , (7) ∂ x2 where μ = n−0 /n+0 is the ratio between the equilibrium number densities of negative and positive ions. Here, φ is normalized by 2k B T F e /e, space is normalized by c s+ /ω p + and time by the inverse



1 2 positive ion plasma frequency ω− p + = m+ /4π n+0 e . We next use the reductive perturbation method to derive a KdV equation with higher-order corrections from Eqs. (1)–(7). We concentrate on the one-dimensional case. The stretched coordinates

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ξ = ε 1/2 (x − λt ), and τ = ε 3/2 t are introduced, where λ is the phase velocity of the QIAWs and ε is a small but nonzero parameter measuring the strength of nonlinearity. The depended variables are expanded as

n j = 1 + εn j1 + ε 2 n j2 + ε 3n j3 + · · · , v e = v 0 + ε v e1 + ε 2 v e2 + ε 3 v e3 + · · · , v ± = 0 + ε v ±1 + ε 2 v ±2 + ε 3 v ±3 + · · · ,

φ = 0 + ε φ1 + ε 2 φ2 + ε 3 φ3 + · · · .

(8)

Substituting these expansions into Eqs. (1)–(7) and collecting terms for different powers of ε , we have the lowest order n+1 = φ1 /(λ2 − σ+ ), n−1 = −μ− φ1 /(λ2 − σ− ), and ne1 = −σe φ1 /[(λ − v 0 )2 − σe ]. By introducing the lowest-order variables into Poisson’s equation, we obtain the dispersion relation

(1 − μ)σe μμ− 1 + + 2 = 0. 2 2 λ − σ− (λ − v 0 ) γ1 − σe λ − σ+

(9)

The dispersion relation can be written in a polynomial form which is quartic in λ, from which we obtain four QIAW modes. The coefficients of the cubic term and linear term will be zero when the beam velocity is zero, in which case one instead obtains two QIAW modes. In the next higher order of ε 5/2 , eliminating the secondorder quantities, we have

∂φ1 ∂φ1 ∂ 3 φ1 + A φ1 =0 +B ∂τ ∂ξ ∂ξ 3

(10)

where the coefficients of nonlinear term and dispersive terms of Eq. (10) are defined as:

A= B=

(3λ2 + σ+ )χ+3 − μμ2− (3λ2 + σ− )χ−3 − δe (1 − μ)χe3 , (11) δ0 χ 0 4χ02 − (1 − μ)σe H e2 cn4 γ0−1 χe2 4 δ0

(12)

,

where the parameter are χ− = ( S 2 γ1 − σe )(λ2 − σ+ ), χ+ = ( S 2 γ1 − σe )(λ2 − σ− ), χe = (λ2 − σ− )(λ2 − σ+ ), χ0 = (λ2 − σ− )( S 2 γ1 − σe )(λ2 − σ+ ), and δe = [σe2 S 2 (2γ0 γ1 + α ) + γ0 σe3 + γ2 σe2 S ( S 2 γ1 − 2σe )]γ0−1 , and δ0 = 2μμ− λχ−2 + 2γ1 σe S (1 − μ)χe2 + 2 2λχ+ . The other parameters are S = λ − v 0 , γ0 = 1 − v 20 /2cn2 , γ1 = 1 − 3v 20 /2cn2 and γ2 = v 0 /cn2 , and α = −(3γ22 v 0 − 4γ2 ) S + γ0 γ1 .

Eq. (10) govern small but finite amplitude relativistic quantum ion acoustic waves in a plasma with two polarity ions, and electron beams. We will now consider some higher-order corrections to the QIAWs. In a similar way as above, the second-order quantities are given as

n +2 = X + n −2 = X − ne2 = X e

2

φ12 2

φ12 2

v + 2 = κ+ v − 2 = κ− v e2 = κe

φ12

2

φ12 2

2

∂ 2 φ1 , ∂ξ 2

+ Y − φ2 + Z −

∂ 2 φ1 , ∂ξ 2

+ Y e φ2 + Z e

φ12

φ12

+ Y + φ2 + Z +

∂ 2 φ1 , ∂ξ 2

+ Y + λφ2 + ( Z + λ + Y + B )

∂ 2 φ1 , ∂ξ 2

+ Y − λφ2 + ( Z − λ + Y − B )

∂ 2 φ1 , ∂ξ 2

+ Y e S φ2 + ( Z e S + Y e B )

∂ 2 φ1 , ∂ξ 2

(13)

3

2 where the coefficients are κ+ = X + λ + Y + A − 2Y + λ , κ− = X − λ + 2 2 Y − A + 2Y − λ, and κe = X e S + Y e A + 2Y e S, where other parameters X j , Y j , Z j are X + = [−2 A λ(λ2 − σ+ ) + 3λ2 + σ+ ]/(λ2 − σ+ )3 , X − = [2 A μ− λ(λ2 − σ− ) + 3μ2− λ2 + σ− μ2− ]/(λ2 − σ− )3 ,

X e = [2 A γ0 γ1 σe S ( S 2 γ1 − σe ) + δ0 γ0−1 ]/( S 2 γ1 − σe )3 γ0 , Y + = 1/(λ2 − σ+ ), Y − = −μ− /(λ2 − σ− ), Y e = −σe /( S 2 γ1 − σe ), Z + = −8B λ/4(λ2 − σ+ )2 , Z − = 8B μ− λ/4(λ2 − σ− )2 , and Z e = (8B γ0 γ1 σe S + σe H e2 cn4 )/4( S 2 γ1 − σe )2 γ0 . At the higher order of ε7/2 , eliminating the third-order quantities n+3 , n−3 , ne3 , v +3 , v −3 , and v e3 with the help of first-order quantities and Eqs. (13), we obtain the inhomogeneous equation for higher-order corrections as

∂φ2 ∂φ1 φ2 ∂ 3 φ2 ∂φ1 ∂ 3 φ1 +A = L φ12 +B + M φ1 3 ∂τ ∂ξ ∂ξ ∂ξ ∂ξ 3  2 ∂ 5 φ1 ∂ ∂φ1 +N + P , ∂ξ ∂ξ ∂ξ 5

(14)

where the coefficients of the higher-order inhomogeneous equations (14) are L = E χ02 /δ0 , M = F χ02 /δ0 , N = G χ02 /δ0 , and P = K χ02 /δ0 . The parameters E, F , G, and K are given in Appendix A. Since the coefficients of the KdV equation (10) and inhomogeneous equation (14) are functions of the streaming velocity of the electrons, quantum parameters, as well as the ratio of number density of negative ions to that of positive ions, the amplitudes and widths of the solitons will also depend on these parameters. In the next section we will derive analytic solution for QIA solitons and investigate the effects of streaming velocity, negative ions and quantum effect on the shapes of the solitons. 3. Solutions and discussions In order to investigate the quantum effects on the shapes of the QIA solitons and the higher-order corrections to the QIA solitons, we analyze the dispersion relation of the QIAWs numerically. The dispersion relations equation is quartic in phase velocity and can give rise to four QIAW modes propagating with different phase velocities. As illustrated in Fig. 1, however, our numerical results give only one mode with physical meaning propagating in the positive direction and one mode propagating in negative direction, which is due to the fact that we only consider streaming electrons. If thermal electron was also present in our system, the right-hand side of the dispersion relations (9) will not be zero, which has four modes. Here we want to consider the streaming effect on the QIAWs. Accordingly we give the phase velocity versus the streaming velocity in Fig. 1. The results show that the phase velocity of the QIAWs in negative quantum plasmas increases with the increase of the ratio of the number densities of negative and positive ions. In Fig. 1, we used the parameters n+0 = 1023 cm−3 , me = 9.109 × 10−28 g, m+ = m− = 1836me . The dispersion curves in Fig. 1 correspond to the concentration of negative ions μ = 0.1 (solid line), μ = 0.3 (dashed line), μ = 0.5 (dotted line), and μ = 0.7 (dashed-dotted line), and the quantum parameter H e = 5.4 × 10−7 . From Fig. 1 one can see that the phase velocity decreases with increasing electron beam velocity. The dispersion relation of QIAWs will be destroyed when the beam velocity is larger than a critical value. In Fig. 2, we also used n+0 = 1031 cm−3 , which corresponds to the quantum parameter H e = 0.005. The other parameters are the same as in Fig. 1. The parameters are typical for dense plasmas, such as in pulsar magnetospheres, the atmosphere of neutron stars or white dwarfs [40]. When the beam velocity is small enough, the dispersion relation of the QIAWs in Fig. 2 is similar to that illustrated in Fig. 1. However when the beam velocity is larger than a critical value, the phase velocity of the QIAWs increases sharply with the increase of the beam velocity. Moreover there is no critical beam velocity for

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The parameter δθ was introduced here to cancel the secular term in the right-hand side of the normalized inhomogeneous equation. The normalization was already used in the higher-order corrections to QIAW solitons in dusty pair ions plasmas [36] and in electron–positron–ion plasmas [37]. We look for steady-state so1 = φ 1 (η) lutions of the normalized KdV equation in the form φ with η = ξ − (θ + δθ)τ . Now using the appropriate boundary con1 → 0, ∂ φ 1 /∂ η2 → 0 at η → ∞, the 1 /∂ η → 0, ∂ 2 φ ditions, viz, φ solution of the normalized KdV equation as

1 = φ0 sech2 (Δη), φ

(17)

where φ0 = 3θ/ A is the amplitude and Δ = θ/4B the width of the QIAW soliton. The method of variation of parameters can be used to solve the inhomogeneous equation to find particular solution [42]. Here the series solution method is used to investigate higher-order solution. Accordingly the dressed soliton solution with the higher-order corrections can be written as [43]

Fig. 1. Dispersion relations of QIAWs with the parameters μ = 0.1 (solid line), μ = 0.3 (dashed line), μ = 0.5 (dotted line), μ = 0.7 (dash-dotted line) and the quantum parameters H e = 5.4 × 10−7 .

η) + φ p , = φ 0 sech2 (Δ φ

where the second term in the dressed solutions is the particular solution of the inhomogeneous equation and can be written as [43]

p = φ



15N θ 2 2 A B2



+ −

Fig. 2. Dispersion relations of QIAWs with the parameters μ = 0.1 (solid line), μ = 0.3 (dashed line), μ = 0.5 (dotted line), μ = 0.7 (dash-dotted line) and the quantum parameters H e = 0.005.

QIAWs modes in the ultra-high density plasmas. One also finds that the phase velocity of the QIAWs increases with the increase of the ratio of number density of negative ions to that of positive ions. By using the method of renormalization [41] to derive the solution of the KdV equation (10) and inhomogeneous equation (14), we obtain

1 1  1 ∂φ ∂φ ∂φ ∂ 2φ 1 1 + B + Aφ + δθ = 0, 3 ∂τ ∂ξ ∂ξ ∂ξ

(15)

2 2 1 φ 2 2 ∂φ ∂φ ∂φ ∂ 3φ +A + δθ +B 3 ∂τ ∂ξ ∂ξ ∂ξ 12 = Lφ

1 1 1 ∂φ ∂ 3φ ∂ 5φ ∂ 1 +N +P + Mφ 3 ∂ξ ∂ξ ∂ξ ∂ξ 5



1 ∂φ ∂ξ

2 + δθ

1 ∂φ . ∂ξ (16)

(18)

−

3(2M + P )θ 2

45N θ 2 4 A B2

A2 B

+

+

9( M + P )θ 2 2 A2 B

9L θ 2 A3

−



η) sech2 (Δ

9L θ 2 2 A3



η) sech4 (Δ

(19)

where the amplitude and width of KdV soliton with higher-order 0 = 3(θ + δθ)/ A and Δ  = (θ + δθ)/4B with the corrections are φ parameter δθ = − N θ 2 / B 2 . We next numerically investigate the contribution of the higherorder corrections to the dressed QIA soliton. In Fig. 3, we give the combined effects of electron beam velocity, negative ions, and the quantum effects on the shape of the KdV soliton, the higher-order solution and the dressed soliton by using the dispersion relation of QIAWs illustrated in Fig. 2. The parameters are (a) μ = 0.1 and λ = 1.22, (b) μ = 0.3 and λ = 1.49, (c) μ = 0.5, and λ = 1.89, and (d) μ = 0.7 and λ = 2.58 with v 0 = 20.57, μ− = 1 and θ = 0.3 and the other parameter are the same as that in Fig. 2. From Fig. 3 one can see that the contribution of the higher-order corrections become less important when the ratio of the number density of negative ions to positive ions increases. We also find that the amplitude and width of the KdV solitons, higher-order solutions and dressed solitons all increase with the increase of the ratio μ. Now we discuss the effect of quantum diffraction on the amplitudes of the KdV soliton φ01 , of the higher-order correction φ02 , and of the dressed soliton φ03 , which are illustrated in Fig. 4. We also use the dispersion relation given by Fig. 2 to determine the relationship between phase velocity and streaming velocity. The parameters are (a) μ = 0.1, λ = 1.2, v 0 = 20.6 and (b) μ = 0.7, λ = 2.6, v 0 = 20.6. These parameters are obtained from Fig. 2. Moreover μ− = 1 and θ = 0.3, while the other parameters are the same as that in Fig. 2. In Fig. 4, we illustrate the quantum effect on the amplitude of the QIAW solitons, where we can see that the amplitude of the KdV soliton (dashed line) and dressed soliton (dash-dotted line) decreases with the increase of the quantum diffraction effects. For large quantum parameters H e = 0.006, the amplitude of higher-order correction φ02 is almost equal to that of the KdV soliton φ01 at μ = 0.1. For μ = 0.7 the higher-order correction contribution is comparable to half of KdV soliton at H e = 0.006. Then one can conclude that for dense quantum plasmas the higher-order correction deserves to be considered.

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The combined effects of quantum diffraction, electron beam velocity, negative ions on the shape of the KdV soliton, of higherorder correction and of the dressed soliton of QIAWs were also investigated numerically. The main results are that the contribution of the higher-order correction become small when the ratio of the number density of negative ions to positive ions is large. While the amplitude and width of the KdV solitons, of higher-order solutions and of dressed solitons of QIAW all increase with the increase of the proportion of negative ions. Moreover the amplitudes of the KdV soliton and of dressed soliton decrease with the increase of the quantum diffraction effects. While the higher-order correction become more and more important for large quantum effects relatively. Acknowledgements One of the authors (Y.W.) thanks the support of the National Natural Science Foundation of China (No. 11104012) and the Fundamental Research Funds for the Central Universities (No. FRF-TP09-019A, No. FRF-BR-11-031B). 1 (dashed line) and higher oder correction φ 2 (solid line) Fig. 3. The soliton φ  (dash-dotted line) with the different ratio (a) μ = 0.1, and the dressed solitons φ (b) μ = 0.3, (c) μ = 0.5, and (d) μ = 0.7. The quantum diffraction parameter of positive ions is H e = 0.005.

Appendix A The coefficients appearing in the right-hand side of inhomogeneous equation (19) are given as

(1 − μ) 1 Ee − 2 E +, ( S 2 γ1 − σe )γ0 λ − σ+ (1 − μ) μ 1 F= 2 F− + 2 Fe − 2 F +, λ − σ− ( S γ1 − σe )γ0 λ − σ+ (1 − μ) μ 1 G= 2 G− + 2 Ge − 2 G +, λ − σ− ( S γ1 − σe )γ0 λ − σ+ (1 − μ) μ 1 K= 2 K− + 2 Ke − 2 K+, λ − σ− ( S γ1 − σe )γ0 λ − σ+ E=

μ

λ2 − σ−

E− +

(A.1) (A.2) (A.3) (A.4)

The quantities E −,e,+ , F −,e,+ , G −,e,+ , and K −,e,+ are presented in the following

Fig. 4. The amplitude of the KdV soliton φ01 (dashed line), of higher-order correction φ02 (solid line) and of dressed soliton φ03 (dash-dotted line) versus quantum parameter of electrons. The parameters are (a) μ = 0.1 and (b) μ = 0.7.

9 2 E − = −2 A X − λ − A 2 Y − + AY − λ + X − Y − λ2 2 3 3 2 + 6Y − λ + X − Y − σ− , 2 3 2 E e = Y e A S (γ0 γ1 + α ) − γ2 Y e3 S σe − 3 A X e S γ0 γ1 2

− A 2 Y e γ0 γ1 − 2 AY e2 S γ0 γ1 − 3v 0 AY e2 S 2 γ22 + 3 X e Y e S 2 γ0 γ1 + 3Y e3 S 2 (γ0 γ1 + α ) +

4. Conclusions

+ In this Letter, the nonlinear propagation of QIAWs was presented in quantum plasmas consisting of two polarity ions and mildly relativistic electron beams. By using the reductive perturbation method, we obtained the KdV equation and inhomogeneous equation for higher-order corrections. The dispersion relations were analyzed numerically, where it was found that the quantum effect can change the dispersion relation significantly. The results show that the phase velocity of the QIAWs will decrease with the increase of electron beam velocity for small quantum diffraction effects. While for large quantum diffraction effects the phase velocity of QIAWs will increase with the increase of the beam velocity that is no cut-off value. When the electron beam velocity is small enough, the dispersion relation of the QIAWs with small quantum effect is similar to that with large quantum effects.

(A.5)

3 2

X e Y e γ0 σe +

1 2

X e S γ2 σe +

+ Y e2 γ2 S σe + Y e3 S 2 β −

γ2 2v 0

1 2

3 2

Xe Y e S 2α

Y e A γ2 σe

Y e3 S 2 σe + 4 AY e2 S 2 γ2 ,

9 2 λ + X + Y + λ2 E + = −2 A X + λ − A 2 Y + + AY + 2 3 3 2 − 6Y + λ + X + Y + σ+ , 2

(A.6)

(A.7)

2 F − = −2B X − λ − 2 A B Y − − 2 A Z − λ + 4B Y − λ

+ 3Y − Z − λ2 + Y − Z − σ− , F e = −2B X e S γ

(A.8)

2 0 1 − 2 A B Y e 0 1 − 2B Y e S 0 1

γ

γ γ

γ γ

2

− 2 A Z e S γ0 γ1 + Y e Z e S (2γ0 γ1 + α ) − 3B v 0 Y e2 S 2 γ22 + 4B Y e2 S 2 γ2 + Y e Z e γ0 σe

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AID:22151 /SCO Doctopic: Plasma and fluid physics

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Y. Wang et al. / Physics Letters A ••• (••••) •••–•••

6

+ B Y e2 S (γ0 γ1 + α ) +

H e2 cn4  4



Y e2 − X e ,

(A.9)

2 F + = −2B X + λ − 2 A B Y + − 2 A Z + λ + 4B Y + λ

+ 3Y + Z + λ2 + Y + Z + σ+ ,

(A.10)

2

G − = −2B Z − λ − B Y − , 2

G e = −2B Z e S γ0 γ1 − B Y e γ0 γ1 −

(A.11) H e2 cn4 4

Ze,

G + = −2B Z + λ − B 2 Y + , 3 3 2 K − = −3 A Z − λ − A B Y − + Y − Z − λ2 + Y − Bλ 2 2 1 + Y − Z − σ− , 2 3 3 K e = −3 A Z e S γ0 γ1 − A B Y e γ0 γ1 − H e2 X e 2 8 1 2 1 1 + Y e B S (2γ0 γ1 + α ) + Z e S γ2 σe + B Y e γ2 σe 2 2 2 H e2 cn4 2 1 + Y e Z e γ0 σe + Y e + Y e Z e S 2 (2γ0 γ1 + α ), 4 2 3 3 2 K + = −3 A Z + λ − A B Y + + Y + Z + λ2 + Y + Bλ 2 2 1 + Y + Z + σ+ . 2

[7] [8] [9] [10] [11] [12]

(A.12)

[13] [14] [15]

(A.13)

[16]

(A.14)

(A.15)

(A.16)

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