Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

Physics Letters A 377 (2013) 2097–2104

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Physics Letters A www.elsevier.com/locate/pla

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution Yue-Yue Wang a , Ji-Tao Li b , Chao-Qing Dai a,∗ , Xin-Fen Chen c , Jie-Fang Zhang d a

School of Sciences, Zhejiang A&F University, Lin’an 311300, Zhejiang, China Shangqiu Medical College, Shangqiu 476100, Henan, China c Electronics and Information Engineering Department, Wuxi City College of Vocational Technology, Wuxi 214153, Jiangsu, China d Zhejiang University of Media and Communications, Hangzhou 310018, Zhejiang, China b

a r t i c l e

i n f o

Article history: Received 23 November 2012 Received in revised form 31 May 2013 Accepted 10 June 2013 Available online 13 June 2013 Communicated by F. Porcelli Keywords: Electron acoustic plasma Solitary wave Rogue wave Nonlinear Schrödinger equation

a b s t r a c t In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α . Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Electron acoustic (EA) waves not only occur in space plasmas such as in the Earth’s bow shock [1] and in the auroral magnetosphere [2], but also can be obtained in laboratory experiments [3–5]. For instance, the excitation of EA waves has been obtained in recent experiments with non-neutral plasmas by the non-neutral plasma physics group at University of California at San Diego [5]. The propagation of EA waves in plasmas with two-temperature (cold and hot) electrons has received a great deal of attention because of its vital role in understanding different types of collective processes in laboratory devices as well as in space environments [6,7]. EA waves in such plasmas are typically high frequency waves because its frequency is much higher than the ion plasma frequency. Therefore, ions remain stationary and form a neutralized background. They provide charge neutrality but do not play an essential role in the dynamics. The cool electrons provide the inertia necessary to maintain the electrostatic oscillations, while the restoring force comes from the hot electrons’ pressure. A lot of studies focus on the nonlinear evolution of EA waves in such plasmas. One of the most famous nonlinear EA waves is the electron acoustic solitary wave, which is observed in experiments with pure electron plasmas [8] and in laser-produced plasmas [9]. Many researchers have done excellent job in studying EA solitary waves theoretically and numerically [10–13]. For instance, in Ref. [13], the excitation of long-lived electrostatic solitons were obtained in unmagnetized plasma of electrons and ions by driving the system with an external electric field. The excitation of such acoustic solitons is triggered by the resonant interaction of electrostatic waves and particles that deform the particle distribution function through the generation of trapped particle regions. All these researches consider the electrons follow the Maxwellian distribution, which is believed to be valid universally for the macroscopic ergodic equilibrium systems. Even so, numerous observations of space plasmas are often characterized by a particle distribution function with high energy tail, but such distribution may deviate from Maxwellian distribution [14,15]. The non-equilibrium stationary states exist in the systems with the long-range interactions such as plasma and gravitational systems. Therefore, Maxwellian distribution might be inadequate for the description of these systems. Tsallis [16] consistently extended Boltzmann–Gibbs (BG) thermodynamics by extending the concept of entropy to the nonextensive fields. In these fields, the entropic

*

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0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.06.008

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index q characterizes the degree of nonextensivity of the considered system (q = 1 corresponds to the standard extensive, BG statistics). Indeed, many physical systems, which cannot be correctly explained by using the classical statistical description, can be described by the appropriate framework of nonextensive statistics. For instance, some researchers have discussed details of the wave–particle interaction in the case of plasmas of nonextensive electrons [17–19]. In these papers, the effect of the Landau damping saturation is analyzed as a function of different values of the nonextensive parameter q. It is found that effect of nonlinear wave–particle interaction produces distortion of the particle distribution function. Much more evidence has shown that Tsallis q-entropy and the ensuing nonextensive statistics may be very important for systems endowed with long-range interactions as usually happens in astrophysics and plasma physics. It is proved to provide a convenient frame for the analysis of problems in plasma physics, long-range Hamiltonian systems, gravitational systems, astrophysical environments, and so on [20,21]. For instance, the Tsallis statistics is commonly used in core-halo distributions frequently detected under typical interplanetary conditions [22], and the analysis of turbulence and intermittency phenomena in solar wind plasmas [23]. The presence of the nonextensive electron component has been proved to have a great influence upon the Modulational Instability (MI) of the EA waves and play a vital role in the formation of solitary waves. Meanwhile, a lot of studies have already confirmed the existence of nonthermal electrons through the observation of a variety of astrophysical plasma environments [24,25]. Therefore, we mainly discuss the solitary waves and rogue waves in EA waves containing the population of Tsallis electrons. In recent years, rogue wave also arouses people’s great interest. The rogue wave (or freak wave), which is firstly found in the ocean with amplitude much higher than the average wave crests around it [26], has attracted more and more attention of the researchers. The rogue wave is a short-lived phenomenon which will suddenly appear out of normal waves but with a small probability. It was considered mysterious until direct measurements confirmed its appearance in real life [27]. Indeed, rogue wave can be detected in various nonlinear physical environments, such as optical systems [28,29], Bose–Einstein condensates [30], superfluid helium [31], atmosphere [32], and even the financial market [33]. Recently, more and more researchers pay attention to the rogue wave phenomena in plasmas. For instance, in Ref. [34], authors investigated the nonlinear Langmuir rogue wave in collisionless electron–positron plasmas. Authors [35] presented an investigation for the generation of a dust acoustic rogue wave in a dusty plasma composed of negatively charged dust grains, as well as nonextensive electrons and ions. Ref. [36] showed that solitary and freak waves could propagate in a dusty plasma composed of positive and negative ions, as well as nonextensive electrons. The generation of nonlinear ion acoustic waves in a plasma with nonextensive electrons and positrons was also studied [37]. Linear wave theories have difficulties in explaining such waves [38,39]. In contrast, nonlinear theories are promising [40,41]. Generally speaking, rogue waves represent an extreme sensitivity of the nonlinear system to the initial conditions. These waves may arise from the instability of a certain class of initial conditions that tend to grow exponentially and thus have the possibility of increasing up to very high amplitudes due to MI [42]. Moreover, many researchers confirm that one of the best ways to describe rogue waves mathematically is the rational solutions of the nonlinear Schrödinger (NLS) equation. Besides the theoretical study on rogue wave in plasma, the experimental study has also made great progress. Experimental observations of rogue waves in a multicomponent unmagnetized plasma have been reported [43]. As the nonlinear features of EA wave have not been understood thoroughly, it is meaningful to investigate not only the EA solitary wave but also the EA rogue wave in plasmas with nonthermal hot electrons featuring the Tsallis distribution. To the best of our knowledge, the EA solitary wave and EA rogue wave in such plasmas has not been investigated yet. Moreover, the rogue wave may cause disaster in ocean, but in optical media, the high amplitude feature of rogue wave can be applied to excite high intensity optical pulses. Thus, it is important and meaningful to study how to control rogue wave in plasmas. In this Letter, we will not only focus on the study of the Modulation Instability (MI) which causes the emergence of the solitary wave and rogue wave, but also investigate the effect of nonextensive parameter q and nonthermal electrons parameter α on the existential region and the amplitude of these nonlinear waves. Then, we give the analytical solitary wave solutions and rogue wave solutions for NLS equation by using the Darboux transformation method [44–46]. Finally, the collision of two solitary waves and rogue waves as well as the control of rogue waves are investigated. 2. Basic equations and derivation of the NLS equation 2.1. Basic equations We consider a collisionless unmagnetized plasma consisting of cold fluid electrons, stationary ions and hot nonthermal electrons featuring Tsallis distribution. The normalized equations are as follows [47]:

⎧ n ∂(nc u c ) ∂ c ⎪ + = 0, ⎪ ⎪ ⎪ ∂ t ∂x ⎪ ⎪ ⎨ ∂ uc ∂φ ∂ uc + uc =δ , ∂ t ∂ x ∂x ⎪ ⎪   ⎪ 2 ⎪ ⎪ ∂ φ 1 1 ⎪ ⎩ , = nc + nh − 1 + δ δ ∂ x2

(1)

where nc (nh ) is the cold(hot) electron number density normalized by its equilibrium value nc0 (nh0 ), u c represents the cold electron fluid velocity normalized by k B T h /δme , where δ = nh0 /nc0 . φ is the electrostatic potential normalized by k B T h /e. The time and distance are





1 2 2 in units of the cold electron plasma period ω− pc = me /4π nc0 e and the hot electron Debye length λ Dh = k B T h /(4π nh0 e ), respectively. The hot electrons are assumed to follow the nonextensive nonthermal velocity distribution function and the normalized hot electron density is given as



nh = 1 + (q − 1)φ

2(qq+−11)



1 + Aφ + B φ2 ,

(2)

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16α q(2q−1)

16α q

where A = − (5q−3)(3q−1)+12α and B = (5q−3)(3q−1)+12α with the nonthermal parameter determining the number of nonthermal electrons α . q stands for the strength of the nonextensivity. The parameter q that underpins the generalized entropy of Tsallis is linked to the underlying dynamics of the system under concern, and provides a measurement of the degree of its correlation and its deviation from the Maxwell–Boltzmann thermodynamic equilibrium. In the extensive limiting case (q → 1), Eq. (2) reduces to the well-known Cairns nonthermal density [48]



nh = 1 − If

4α 1 + 3α

φ+

4α 1 + 3α



φ

2

exp(φ).

(3)

α = 0, Eq. (2) reduces to the nonextensive electron density [49]

q +1 nh = 1 + (q − 1)φ 2(q−1) .

(4)

2.2. Derivation of the NLS equation We introduce independent variables scaled as ξ = (x − ct ), τ = 2 t, where is a very small constant and c is determined by the group velocity. According to the reductive perturbation (RP) method, the dependent variables are expanded as

⎧ ∞ ∞   ⎪ ( j) ⎪ ⎪ nc = 1 +

j ncl (ξ, τ )e i (kx−ωt )l , ⎪ ⎪ ⎪ ⎪ j =1 l=−∞ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ⎨   ( j) uc =

j u cl (ξ, τ )e i (kx−ωt )l , ⎪ ⎪ j =1 l=−∞ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ⎪   ⎪ ( j) ⎪ j ⎪

φl (ξ, τ )e i (kx−ωt )l , φ = ⎪ ⎩ j =1

(5)

l=−∞

where k and ω are real variables representing the carrier wave number and frequency, respectively. Substituting Eq. (2) and Eq. (5) into Eq. (1) and collecting the terms in different powers of , we obtain the first-order ( j = 1) quantities with l = 1 as follows: (1 )

nc1 = −

k 2 δ (1 )

ω

φ1 ,

2

(1 )

u c1 = −

k δ (1 )

ω

(6)

φ1 .

The linear dispersion relation is obtained as









ω2 = 2k2 15q2 − 14q + 3 + 12α / 30q2 − 28q + 6 + 24α k2 + (5q − 3) 3q2 + 2q − 1 − 4α . (1)

(7) (1)

(1)

The constant c is determined by the group velocity, namely, c = ∂ ω/∂ k = ω(1 − ω )/k. Moreover, u 0 = n0 = Φ0 j = 1, l = 0. Based on Eq. (6), we express the second-order ( j = 2) quantities with l = 1 as 2

(2 )

nc1 = −i

δk

ω3



(1 ) (1 )  ∂φ1 ∂φ1 (2 ) 2ck ω , − ikωφ1 − 2 ∂ξ ∂ξ

(2 )

u c1 = −i

δ

ω2



(1 )

= 0 are obtained for

(1 ) 

∂φ ∂φ (2 ) ck 1 − ikωφ1 − ω 1 ∂ξ ∂ξ

.

(8)

The zeroth-harmonic modes, which would appear due to the self-interaction of the modulated carrier waves, are obtained for j = 2, (1) l = 0 in terms of |φ1 |2 as





(2 ) (1 ) 2 nc0 = A 1 φ1  ,





(2 ) (1 ) 2 u c0 = B 1 φ1  ,

 (1) 2 (2 ) φ0 = C 1 φ1  ,

(9)

where A 1 , B 1 , C 1 are defined as follows:

⎧ k2 δ(5q−3)[2δk2 (2ω2 −3)(3q2 +2q−1−4α )+ω2 (3q−1)(q2 −2q−3−12α )] ⎪ ⎪ ⎪ A 1 = 2ω2 [k2 (30q2 −28q+6+24α )−ω2 (ω2 −1)2 (5q−3)(3q2 +2q−1−4α )] , ⎪ ⎨ kδ[2δ ω2 k2 (ω2 −1)(5q−3)(3q2 +2q−1−4α )−8δk4 (15q2 −14q+3+12α )−ω4 (5q−3)(3q−1)(ω2 −1)(q2 −2q−3−12α )] B1 = , 2ω3 [k2 (30q2 −28q+6+24α )−ω2 (ω2 −1)2 (5q−3)(3q2 +2q−1−4α )] ⎪ ⎪ ⎪ 4 2 2 4 2 2 2 ⎪ ⎩ C 1 = − 4k δ(2ω −3)(15q −14q+3+12α )+ω (5q−3)(3q−1)(ω −1) (q −2q−3−12α ) . 2ω2 [k2 (30q2 −28q+6+24α )−ω2 (ω2 −1)2 (5q−3)(3q2 +2q−1−4α )]

(10)

Similarly, the second-harmonic modes ( j = 2, l = 2) arising from the nonlinear self-interaction of the carrier waves are obtained in (1) terms of (φ1 )2 as (2 )



( 1 ) 2

nc2 = A 2 φ1

,

(2 )



( 1 ) 2

u c2 = B 2 φ1

,

( 1 ) 2 (2 ) φ2 = C 2 φ1 ,

(11)

where A 2 , B 2 , C 2 are defined as follows:

⎧ k2 δ[48k4 δ(15q2 −14q+3+12α )+6k2 δ(5q−3)(3q2 +2q−1−4α )−ω2 (5q−3)(3q−1)(q2 −2q−3−12α )] ⎪ A2 = , ⎪ ⎪ 4ω2 [2k2 (4ω2 −1)(15q2 −14q+3+12α )+ω2 (5q−3)(3q2 +2q−1−4α )] ⎪ ⎨ 4 2 2 2 2 2 4 2 kδ[8k δ(2ω +1)(15q −14q+3+12α )+2k δ ω (5q−3)(3q +2q−1−4α )−ω (5q−3)(3q−1)(q −2q−3−12α )] B2 = , 4ω3 [2k2 (4ω2 −1)(15q2 −14q+3+12α )+ω2 (5q−3)(3q2 +2q−1−4α )] ⎪ ⎪ ⎪ 4 2 4 2 ⎪ −12k δ(15q −14q+3+12α )+ω (5q−3)(3q−1)(q −2q−3−12α ) ⎩ C 2 = 4ω2 [2k2 (4ω2 −1)(15q2 −14q+3+12α )+ω2 (5q−3)(3q2 +2q−1−4α )] .

(12)

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Fig. 1. (Color online.) (a) The variational curved surface of the frequency ω with the wave number k and nonextensive parameter q, where δ = 2, α = 0.3. (c) The variational curved surface of the frequency ω with the wave number k and nonthermal parameter α , where δ = 2, q = 1.2. (b) and (d) are the contour figures of (a) and (c). (e) and (f) are the contour figures of the variational curved surface of the phase speed v p with the wave number k and nonextensive parameter q, where (e) q < 0.6 and (f) q > 0.6 and δ = 2, α = 0.3. (g) is the contour figure of the variational curved surface of the phase speed v p with the wave number k and nonthermal parameter α , where δ = 2, q = 1.2. The value increases as the color turns from blue to red.

Finally, substituting all the previous derived expressions (Eqs. (6)–(12)) into the components for j = 3, l = 1 of the reduced equations, we will get the following NLS equation

i

∂Φ ∂ 2Φ + P 2 + Q |Φ|2 Φ = 0, ∂τ ∂ξ (1)

where Φ stands for φ1 , P =

3ω3 (ω2 −1) , 2k2

(13) Q = (5q − 3)(3q − 1)ω3 [(−4q2 + 8q + 12 + 48α )(C 1 + C 2 ) + (q + 1)(3q2 − 14q + 60α +

15)]/[32k (15q − 14q + 3 + 12α )] − ω( A 1 + A 2 )/2 − ( B 1 + B 2 )k, in which A 1 , B 1 , C 1 , A 2 , B 2 , C 2 have been mentioned before. Figs. 1(a)–(d) show that the dispersion curve of the EA waves is affected by the nonthermal parameter α and entropic index q. The dispersion relation ω2 is defined in Eq. (7). Figs. 1(a) and 1(b) show that there is a cut-off point q = 0.6 that divides the dispersion curve of the EA waves with the variation of q. ω increases as q increases when q < 0.6, but decreases as q increases when q > 0.6. Figs. 1(c) and 1(d) tell us that ω increases as α increases. Figs. 1(e)–(g) indicate that the phase speed v p = ω/k of the EA waves is affected by nonextensive index q and the nonthermal parameter α . Figs. 1(e) and 1(f) display that v p increases as q increases when q < 0.6, but v p decreases as q increases when q > 0.6. Fig. 1(g) indicates that the phase speed v p of the EA waves increases as α increases. 2

2

3. Discussion The Modulation Instability (MI) of the EA waves can be studied by considering a small perturbation δΦ . We consider the direction 2 of DAW is in the direction of pump carrier for the sake of simplicity. For this purpose we set Φ = (Φ0 + δΦ)e i Q |Φ0 | τ , where Φ0 is the amplitude of the pump carrier which is much larger than the perturbation, i.e., |Φ0 |  |δΦ|. After linearizing Eq. (13), we obtain the governing equation for the small perturbation δΦ :

i



∂δΦ ∂ 2 δΦ +P + Q |Φ0 |2 δΦ + δΦ ∗ = 0, 2 ∂τ ∂ξ

(14)

where δΦ ∗ is the conjugate representation of δΦ . By introducing the transformation δΦ = U + i V , where (U , V ) = (U 0 , V 0 ) exp(i K ξ − i Ω τ ), we obtain the following nonlinear dispersion relation

  Ω 2 = P 2 K 2 K 2 − 2Q Φ02 / P .

(15)

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Fig. 2. (Color online.) (a) The variational curved surface of the MI growth rate Γ with the modulation wave number K and nonextensive parameter q when α = 0.6. (c) The variational curved surface of Γ with K and nonthermal parameter α when q = 0.8. (b) and (d) are the contour figures of (a) and (c). Other parameters are k = 0.5, δ = 2, Φ0 = 1.

Clearly, if P Q < 0, the EA waves are stable in the presence of small perturbation, since Ω is always a real number. On the other hand, when P Q > 0, the MI would set in whenΩ becomes imaginary. This happens when the modulation wave number K of external perturbation is less than the critical value K c = 2Q |Φ0 |2 / P . Furthermore, the MI growth rate is given by

Γ = Im Ω =





P 2 K 2 K c2 − K 2 .

(16)



Obviously, the growth rate reaches its maximum value Γmax = | Q ||Φ0 |2 for K = K c2 /2 when K < K c and P / Q > 0. Fig. 2 shows that the entropic index q and the nonthermal parameter α will influence the MI growth rate Γ . The value of Γ in region 0.6 < q < 0.8 is larger than that in region 0.4 < q < 0.6, and Γ first increases and then decreases in each region (see Figs. 2(a) and 2(b)). In other regions, Modulation Instability does not occur. Figs. 2(c) and 2(d) show that Γ increases as the nonthermal nature of the electrons becomes important. As it is known to all, the MI is related to the emergence of the localized envelope structures (such as bright solitary wave and rogue wave). According to the expressions for P and Q , it is obvious that the sign of the P / Q depends on the wave number k, nonthermal parameter α and entropic index q. For the unstable wave packet ( P / Q > 0), the exponentially growing perturbation cannot last for a long time, since Eq. (14) is derived when δΦ  Φ0 . When the amplitude of the perturbation is comparable to that of the EA waves, the nonlinear structures such as bright solitary wave or rogue wave will emerge. On the other hand, the dark solitary wave, which are dips characterized by a phase jump at the position of the dip, can also appear when P / Q < 0. Thus, the existential regions of different solitary wave for different values of q and α are shown in Fig. 3. The bright solitary wave and rogue wave can only exist in the white regions ( P / Q > 0) while the dark solitary wave can only exist in the dark region ( P / Q < 0). Fig. 3(a) indicates that only the dark solitary wave can exist for small values of k when electrons are q-distributed without any nonthermal electrons, that is when a = 0. Figs. 3(b) and 3(c) show that the existential regions for the bright solitary wave and rogue wave enlarge as α increases, that means such waves are more likely to occur as the nonthermal nature of the electrons becomes important. Note that in the previously published results [17–19], basing on the framework of kinetic Vlasov theory by the Vlasov simulations, authors discussed that the particle trapping (and the consequent nonlinear deformation of the particle velocity distribution) played a crucial role in the excitation of acoustic solitary waves in plasmas. However, in this Letter, we discuss the excitation of electron acoustic solitary waves in the framework of fluid theory. Since the effect of wave–particle interaction that produces distortion of the particle distribution function is not considered in this Letter, we will discuss this topic in the future. By employing the DT, we can obtain exact single solitary wave solution for Eq. (13) as follows:



2P

Φ = β1

Q



sech(β1 ξ − 2P v 1 β1 τ − θ10 ) exp i





ϕ10 + β12 − v 21 P τ + v 1 ξ ,

(17)

where β1 , v 1 , θ10 , ϕ10 are arbitrary constants. We can study the interaction of two solitary waves through the double solitary wave solution, which is obtained by employing DT as follows:



Φ=

2P [a1 cosh θ2 e i ϕ1 + a2 cosh θ1 e i ϕ2 + ia3 (sinh θ2 e i ϕ1 − sinh θ1 e i ϕ2 )] Q

b1 cosh(θ1 + θ2 ) + b2 cosh(θ1 − θ2 ) + b3 cos(ϕ1 − ϕ2 )

where θk = βk ξ − 2P v k βk τ − θk0 , a3 = β1 β2 ( v 1 − v 2 ), b1 =

(β1 −β2 )2 4

ϕk = ϕk0 + (βk2 − v k2 ) P τ + v k ξ (k = 1, 2), and a1 = +

( v 1 − v 2 )2 4

, b2 =

(β1 +β2 )2 4

+

( v 1 − v 2 )2 4

(18)

, β1 2

[β12 − β22 + ( v 1 − v 2 )2 ], a2 =

β2 2

[β22 − β12 + ( v 1 − v 2 )2 ],

, b3 = −β1 β2 , in which βk , v k , θk0 , ϕk0 are arbitrary constants.

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Fig. 3. Region for P / Q > 0 (white region) and P / Q < 0 (black region) as function of the wave number k and q when (a)

α = 0, (b) α = 0.3, and (c) α = 0.6, where δ = 2.

Fig. 4. (Color online.) (a) The evolution figures for the bright solitary wave when k = 0.5, α = 0.5, δ = 2, q = 0.5, θ10 = 0, β1 = 1, v 1 = 1. (b) Interaction of two bright solitary waves when α = 0.5, k = 0.5, θ10 = 0, θ20 = 0, ϕ10 = 0, ϕ20 = 0, β1 = −1.05, β2 = 1, v 1 = 1, v 2 = 1. (c) Interaction of two bright solitary waves when α = 0.5, k = 0.5, θ10 = 5, θ20 = 5, ϕ10 = 0.5, ϕ20 = 0.5, β1 = −1, β2 = 1.01, v 1 = −5, v 2 = 5.

Fig. 4(a) shows the amplitude of bright solitary wave keeps invariant as time goes on. The interactions of two solitary waves are investigated in Figs. 4(b) and 4(c). We find that different types of interactions of the two bright solitary waves will occur due to different initial velocities. Fig. 4(b) shows that two bright solitary waves propagate along their own paths without influencing each other as time goes on. Fig. 4(c) shows that two bright solitary waves attract each other and propagate towards each other. When the collision happens, the amplitude will increase and forms a sharp lump. After that, two solitary waves separate from each other. One can also obtain the first-order (n = 1) and second-order (n = 2) rogue wave solutions (rational solutions) for Eq. (13) by employing the Darboux transformation as follows:



P

n

Φ = (−1)

 1−

Q

(G n + i P (τ − τ0 ) H n )



Fn

  exp i P 1 −

v2 2

  ξv (τ − τ0 ) + i √ ,

(19)

2

ξ

ξ

where G 1 = 4, H 1 = 8, F 1 = 1 + 4( P τ − P τ0 )2 + 4[ √ − v ( P τ − P τ0 )]2 for single rogue wave solution, and G 2 = {[ √ − v P (τ − τ0 )]2 + P 2 (τ − 2 ξ 3 3 3 2 2 2 2 0 ) + 4 }{[ √ − v P ( − 0 )] + 5P ( − 0 ) + 4 } − 4 , 2 ξ ξ F 2 = 13 {[ √ − v P ( − 0 )]2 + P 2 ( − 0 )2 }3 + 14 {[ √ 2 2

τ

τ

τ

τ

τ

τ

τ

τ

τ

2

ξ

ξ

H 2 = P 2 (τ − τ0 )2 − 3[ √ − v P (τ − τ0 )]2 + 2{[ √ − v P (τ − τ0 )]2 + P 2 (τ − τ0 )2 }2 − 15 , 8 2

− v P (τ − τ0 )]2 − 3P 2 (τ − τ0 )2 }2 +

2

9 √ξ [ 16 2

− v P (τ − τ0 )]2 +

33 16

P 2 (τ − τ0 )2 +

3 64

for

double rogue wave solution. In previous works [34–36], the rogue wave is always excited at the initial time τ = 0. However, in our present work, the rogue wave can be excited at a predetermined time because there is a time center (τ0 ) in the rogue solutions. When the time center (τ0 ) increases, the rogue wave can be excited in a later time, that is, the excitation of the rogue wave can be postponed. Fig. 5(a) shows the shape of rogue wave and Fig. 5(b) illustrates the influence of nonextensive parameter q and nonthermal parameter α on the amplitude maximum (|Φmax |) of the rogue wave. It indicates that for small values of q, the rogue wave is much higher than that for large values of q, which means the electron nonextensivity may not only influence the existential region of the rogue wave, but also affect the amplitude of the rogue wave. As the electron nonextensivity increases to a certain value, the inhibition of the amplitude of the rogue wave can be easily implemented. It is necessary to note the rogue wave cannot exist in the vacancy of the line as the MI does not occur. The amplitude of the rogue wave also increases as α increases for the fixed q. It suggests that the quantities of the nonthermal electrons also influence the amplitude of the rogue wave. The amplitude of the rogue wave will increase when the numbers of the nonthermal electrons increase. Fig. 5(d) describes the interaction of two rogue waves. When they collide with each other, the amplitude will increase to a high value if the nonextensive character of the nonthermal electrons is weak. As the number of the nonthermal electrons decreases and the nonthermal electrons nonextensivity becomes more significant, the amplitude of the rogue wave decreases rapidly. Compared with Figs. 5(a) and 5(d), we can see the excitation of rogue wave and interaction of two rogue waves have been postponed in Figs. 5(c) and 5(f), respectively. Since the rogue wave may cause disaster in ocean but become useful in optical media, the control of rogue wave in a physical system is a hot topic for research currently. It is necessary to study the way to control the occurrence of rogue wave in plasmas too. Through the above research, it may be a feasible scheme to excite, postpone and inhibit the occurrence of the rogue waves in plasmas by adjusting the

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Fig. 5. (Color online.) (a) and (d) The evolution figures for the rogue wave and the interactions of two rogue waves, where k = 0.5, α = 0.5, δ = 2, v = 0.1, q = 0.5, τ0 = 3. (b) and (e) The variational curve of |Φmax | as function of q. (c) and (f) Delay excitation of the rogue wave in Fig. 5(a) and interactions of two rogue waves in Fig. 5(d) where τ0 = 10.

parameters, such as the time center τ0 , nonextensive parameter q and nonthermal parameter α . By this way, we can successfully avoid rogue waves or generate highly energetic pulses in plasmas as we wish. It may have an instructive significance for us in controlling rogue waves in plasmas. 4. Conclusion In this Letter, the electron acoustic (EA) waves in a collisionless plasma consisting of stationary ions, cold fluid electrons and hot nonthermal electrons featuring the Tsallis distribution are studied. A nonlinear Schrödinger (NLS) equation is obtained and the Modulation Instability (MI) of the EA waves is studied. The solitary wave and rogue wave are proved to exist in such plasmas and the existential regions for different type of solitary wave and rogue wave solutions are given. The effects of nonthermal parameter α and the nonextensive parameter q on the evolution of the EA waves are also analyzed in great detail. Moreover, the collisions between two solitary waves and the postpone of exciting the rogue waves are studied. These results obtained in this Letter well supplement our comprehension for excitation of solitary waves and control of rogue waves in plasmas. Moreover, these results may have potential values in understanding the nonlinear features of the EA waves in laboratory plasmas and space plasmas, such as in the Earth’s bow shock and in the auroral magnetosphere and so on. Acknowledgements This work has been supported by the Scientific Research Fund of Zhejiang Provincial Education Department (Grant No. Y201225803), the Scientific Research and Development Fund of Zhejiang A&F University (Grant No. 2012FK002), the National Natural Science Foundation of China (Grant Nos. 11072219, 11005092 and 61205006), the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y13F050037). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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