New Homoclinic and Heteroclinic Solutions for Zakharov System

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1 Commun. Theor. Phys. 58 (2012) Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department of Mathematics, Kunming University of Science and Technology, Kunming , China 2 School of Mathematics and Physics, Yunnan University, Kunming , China 3 College of Mathematics and Information Science, Qujing Normal University University, Qujing , China (Received April 16, 2012; revised manuscript received September 5, 2012) Abstract A new type of homoclinic and heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreover, the homoclinic and heteroclinic structure with local oscillation and mechanical feature different from homoclinic and heterocliunic solutions are investigated. Result shows complexity of dynamics for complex nonlinear evolution system. Moreover, the similarities and differences between homoclinic (heteroclinic) breather and homoclinic (heteroclinic) tube are exhibited. These results show that the diversity of the structures of homoclinic and heteroclinic solutions. PACS numbers: a, Jr, m Key words: homoclinic wave, heteroclinic wave, breather type, homoclinic test, Zakharov system 1 Introduction It is well known that the existence of homoclinic and heteroclinic orbits is very important to study the chaotic behavior of partial differential equation. Many methods are developed for proving the existence of homoclinic orbits of perturbed soliton equation, such as nonlinear Schrödinger equation, [1] Sine Gordon equation, [2] DS equation, [3 4] Boussinesq equation, [5] Zakharov system. [6 8] Recently, homoclinic wave solutions with different mechanical features for Sine Gordon equation were investigated. [9] In this work, we focus on Zakharov system ie t + E xx = NE, N tt N xx = ( E 2 ) xx, (1) where E is the envelope of the high-frequency electric field, and N is the plasma density measured from its equilibrium value. It describes the interaction between highfrequency and low-frequency waves which was firstly derived by Zakharov. [10] Zakharov system is non-integrable. It has been extensively studied in various aspect. [6 11] The explicit homoclinic and heteroclinic tube solutions with periodic boundary conditions and even constraints were investigated. [8,11] As we know that the two-soliton method [12 14] and extended homoclinic test method [15 19] are two effective methods for seeking two-soliton and periodic soliton of nonlinear evolution equation. This work applies both of these methods to (1+1)-dimensional Zakharov system, new type of homoclinic and heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions are obtained. Moreover, the homoclinic and heteroclinic solutions structures with local oscillation and mechanical feature of solutions are investigated. At last, the similarities and differences between homoclinic (heteroclinic) breather and homoclinic (heteroclinic) tube [8,11] are exhibited. Obviously the plane wave E = a e i (bx ct+θ), N = N 0, is the solution of Eq. (1) with the relation N 0 = c b 2. By the dependent variable transformation E(x, t) = g(x, t) f(x, t), N = A 2(lnf(x, t)) xx, (2) where A is a constant and g(x, t) and f(x, t) are complex and real functions, respectively. Then, Zakharov system may be rewritten as the following coupled bilinear differential equations for f and g: (D 2 t D2 x B)f f + g g = 0, (i D t + Dx 2 A)g f = 0, (3) where the Hirota bilinear operator D [13] is defined by (n, m 0) ( Dx m Dn t f(x, t) g(x, t) = x ) m ( x t ) n[f(x, t)g(x t, t )] x =x,t =t. Now we apply extended homoclinic test approach to Eq. (1). It has been proved, complex constant (E 0, N 0 ) = (e iθ0, 0) is a hyperbolic fixed point. [11] We now suppose that the solution of Eqs. (3) is in the form f(x, t) = e k2x l2t + δ 1 cos(k 1 x + l 1 t) + δ 2 e k2x+l2t, Supported by the Natural Science Foundation of China under Grant No Corresponding author, zhddai2004@yahoo.com.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 750 Communications in Theoretical Physics Vol. 58 g(x, t) = e iθ (e k2x l2t + δ 3 cos(k 1 x + l 1 t) + δ 4 e k2x+l2t ), (4) where δ i (i = 1, 2, 3, 4), k i, l i (i = 1, 2) are complex. Substitution of (4) into (3) leads to the following relations among these constants where A = 0, δ 3 = (4 k2 1 k2 2 + l2 1 )δ 1 (2 ik 1 k 2 l 1 ) 2, δ 2 = δ2 1 (4 k2 1 k2 2 + l2 1 ) 4l1 2 4, l 1 = ± k 1, k4 2 B = 1, l 2 = l 1(k2 2 k2 1 ), δ 4 = δ 2(2 k 1 k 2 il 1 ) 2 2k 1 k 2 (2 k 1 k 2 + il 1 ) 2, k 2 = ± l 2, (5) = and δ 1, k 1 are arbitrary constants. Combining (4) with (5), we obtain the following solution E(x, t) = e iθ 2 δ 4 cosh(η + ln( δ 4 )) + δ 3 cos(ξ) 2 δ 2 cosh(η + ln( δ 2 )) + δ 1 cos(ξ), 4 k 2 1 l2 2 7 l l4 2 + k2 1 4 l k2 1 l2 2 + l2 2 + k2 1 N(x, t) = 2(2 δ 2 k 2 2 cosh(η + ln( δ 2 )) δ 1 k 2 1 cos(ξ)) 2 δ 2 cosh(η + ln( δ 2 )) + δ 1 cos(ξ) + 2(2 δ 2 k 2 sinh(η + ln( δ 2 )) δ 1 k 1 sin(ξ)) 2 (2 δ 2 cosh(η + ln( δ 2 )) + δ 1 cos(ξ)) 2, (6) where ξ = k 1 x + l 1 t, η = k 2 x + l 2 t and k 1, k 2, l 1, l 2, δ i (i = 2, 3, 4) are given by (5). The solution (E, N) represented by (6) shows a new type of heteroclinic solution, which is a heteroclinic breather heteroclinic to the two different saddle points (e i(θ+θ0), 0) and (e i (θ θ0), 0) of Eq. (1) when t ±, and meanwhile contains a periodic wave whose amplitude periodically oscillates with the evolution of time, i.e. where (E, N) (e i( θ+θ0), 0), as t + ; (E, N) (e i (θ θ0), 0), as t, e i θ0 = 2 k 1k 2 il 1, then θ 0 = arctan 4k 1k 2 l 1 2 k 1 k 2 + il 1 4k1 2k2 2. l2 1 The heteroclinic breather wave shows the elastic interaction between a periodic wave and a heteroclinic wave with the different speed and along the line of different propagation direction. This is a new phenomenon (Fig. 1). Fig. 1 Heteroclinic breather wave behavior of Re(E(x)) and N(x) for the solution (6) in Zakharov system. Now by using two-soliton method we assume the solution with complex wave numbers and frequencies for Eq. (3) has following form: [12] f(x, t) = 1 + e ξ1 + e ξ2 + M e ξ1+ξ2, g(x, t) = a e i(bx ct) (1 + e ξ1+iθ1 + e ξ2+iθ2 + M e ξ1+ξ2+i(θ1+θ2) ), (7) where ξ i = k i x + l i t (i = 1, 2) and θ i, k i, l i (i = 1, 2) are complex parameters. Substitution of (7) into (3) leads to the following relations among these parameters A = N 0, B = a 2, sin 2( θ ) i = l2 i ( k2 i 1 2 a 2, tan i) 2 θ ki 2 =, l i + 2 bk i

3 No. 5 Communications in Theoretical Physics 751 M = δ2 3 δ1 2 2 a 2 sin 2 Θ 1 δ4 2 δ2 2 2a2 sin 2, b = δ2 1 cosθ 1 + δ 3 sin Θ 1 δ 4 M sin Θ 2 + δ2m 2 cosθ 2, (8) Θ 2 2(δ 1 sin Θ 1 + δ 2 M sin Θ 2 ) where δ 1 = k 1 k 2, δ 2 = k 1 + k 2, δ 3 = l 2 l 1, δ 4 = l 2 + l 1, Θ 1 = (θ 1 θ 2 )/2, Θ 2 = (θ 1 + θ 2 )/2. Taking wave numbers and frequencies are complex, respectively, k 1 = k 2 = a + d i, l 1 = l 2 = m + n i, θ 1 = θ 2 = φ + ψ i, (9) where a, d, m, n, φ, ψ are real. Then the relations among these parameters become the following form: A = N 0, B = a 2, sin 2( θ ) i = l2 i ( k2 i 1 2 a 2, tan i) 2 θ ki 2 =, l i + 2bk i M = 2n2 2d 2 a 2 sin 2 φ 2m 2 2a 2 + a 2 sinh 2 ψ, Combining (7) with (10), we obtain the homoclinic solution E(x, t) = a e i(ϕ+φ) M cosh(ξ + iφ) + cos(η + iψ) M cosh(ξ) + cos(η), b = 2a2 M cosφ 2d 2 coshψ n sinhψ mm sin φ 2(aM sin φ d sinhψ). (10) N(x, t) = N 0 2(a2 M d 2 + M(δ cosh(ξ)cos(η) + 2 adsinh(ξ)sin(η))) ( M cosh(ξ) + cos(η)) 2, (11) where ϕ = bx ct, ξ = ax + mt + (1/2)lnM, η = dx + nt, δ = a 2 d 2. The solution (E, N) represented by (11) shows a new type of homoclinic solution, which is a homoclinic breather wave homoclinic to a fixed periodic wave (a e i(ϕ+2φ), c b 2 ) and (a e i ϕ, c b 2 ) of Eq. (1) when t ±, and meanwhile contains a periodic wave whose amplitude periodically oscillates with the evolution of time, i.e. (E, N) (a e i(ϕ+2φ), c b 2 ), as t + ; (E, N) (a e i ϕ, c b 2 ), as t. Note the solution (E, N) of Zakharov system is a three-wave solution containing two periodic wave and a homoclinic wave, which have the different speed c, n and m. It shows the elastic interaction between a periodic wave and a homoclinic wave with different speed and along the line of different propagation direction. This is a new type of homoclinic solution up to now (Fig. 2). Fig. 2 Homoclinic breather wave behavior of Re(E(x)) and N(x) for the solution (11) in Zakharov system. These are new homoclinic and heteroclinic breather solutions for Zakharov system which are different from homoclinic tube and heteroclinic tube solutions obtained in Refs. [8,11]. In Ref. [8], the authors obtained homoclinic tube solution as follows E(x, t) = e iat 1 + 2b 1 cos(px)e kt+γ + b 3 e 2kt+2γ 1 + 2b 4 cos(px)e kt+γ + b 5 e 2kt+2γ, N(x, t) = a + 8p2 + 4b 2 4 p2 cos(px)(e kt γ + b 5 e kt+γ ) (e kt γ + 2b 4 cos(px) + b 5 e kt+γ ) 2, (12) where a, b 1, b 3, b 4, b 5, k, p satisfy (8) in Ref. [8]. The homoclinic tube solution (E, N) is homoclinic to (e iat, a), which is just the hyperbolic fixed point. [8] But the homoclinic wave (11) homoclinics to a fixed periodic wave (a e i ϕ, c b 2 ) when t ±. What is more, the homoclinic tube solution (E, N) is the local stability by the linearized stability analysis and computer simulation (Fig. 3(b)). However, the homoclinic wave solution (11) is different from homoclinic tube solution via computer simulation which is the localized oscillation (Fig. 3(a)). So the homoclinic breather solution (11) is a new type of homoclinic solution.

4 752 Communications in Theoretical Physics Vol. 58 Fig. 3 (a) The local oscillation behavior of E(t) for solution (6); (b) The stability behavior E(t) for solution (13) with x = 10, 20, 30, 40. Fig. 4 (a) The local oscillation behavior of E(t) for solution (11); (b) The stability behavior E(t) for solution (13) with x = 10, 20, 30, 40. In Ref. [11], the authors obtained heteroclinic tube solution as follows E(x, t) = e iθ b 1 cos(px)e kt+γ + b 3 e 2kt+2γ 1 + 2b 4 cos(px)e kt+γ + b 5 e 2kt+2γ, N(x, t) = a + 8p2 + 4b 2 4p 2 cos(px)(e kt γ + b 5 e kt+γ ) (e kt γ + 2b 4 cos(px) + b 5 e kt+γ ) 2, (13) where a, b 1, b 3, b 4, b 5, k, p satisfy (15) in Ref. [11]. Although the heteroclinic tube solution [11] given by (13) and heteroclinic breather solution (6) show that there are heteroclinic to the two different fixed points (e i(θ+θ0), 0) and (e i (θ θ0), 0) of Eq. (1) when t ±, the heteroclinic breather solution (6) is different from heteroclinic tube solution (13). The heteroclinic tube solution (E, N) is the local stability by the linearized stability analysis and computer simulation (Fig. 4(b)). However, the heteroclinic breather solution (6) represents the localized oscillation again and again (Fig. 4(a)). That is, the heteroclinic breather solution (6) is a new type of heteroclinic solution. In summary, applying Hirota bilinear form, two-soliton method and extended homoclinic test approach to Zakharov system, we obtain a new type of homoclinic and heteroclinic solutions. We also investigate the different homoclinic and heteroclinic structure of solutions. These results show the complexity of dynamical behavior and the variety of structure for homoclinic and heteroclinic solutions of Zakharov system. These obtained results show that the diversity of the structures of homoclinic and heteroclinic solutions. Following these ideas in this work, study may be needed further to see whether Eq. (1) has other type of specially spatiotemporal structure of solutions. References [1] Y. Li, J. Nonlinear. Sci. 9 (1999) 363. [2] J. Shatah and C. Zeng, Comm. Pure. Appl. Math. 53 (2000) 283. [3] J. Huang and Z. Dai, Chaos, Solitons & Fractals 35 (2008) 996. [4] Z. Dai and J. Huang, Chin. J. Phys. 43 (2005) 349. [5] Z. Dai, J. Huang, M. Jiang, and S. Wang, Chaos, Solitons & Fractals 26 (2005) 1189.

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