New Soliton-like Solutions and Multi-soliton Structures for Broer Kaup System with Variable Coefficients

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1 Commun. Theor. Phys. (Beijing, China) 44 (005) pp c International Academic Publishers Vol. 44, No. 5, November 15, 005 New Soliton-like Solutions and Multi-soliton Structures for Broer Kaup System with Variable Coefficients JI Ming-Jun 1, and LÜ Zhuo-Sheng 1 Transportation Management College, Dalian Maritime University, Dalian 11606, China Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing , China (Received March 14, 005) Abstract By using the further extended tanh method [Phys. Lett. A 307 (003) 69; Chaos, Solitons & Fractals 17 (003) 669] to the Broer Kaup system with variable coefficients, abundant new soliton-like solutions and multi-solitonlike solutions are derived. Based on the derived multi-soliton-like solutions which contain arbitrary functions, some interesting multi-soliton structures are revealed. PACS numbers: 0.03.Jr, Yv Key words: further extended tanh method, soliton-like solution 1 Introduction Modern soliton theory is widely applied in almost all of the physics fields such as fluid mechanics, plasma physics, nonlinear optics, condensed matter physics, etc. [1] Particularly, the direct search for soliton-like structures for the soliton systems becomes more and more attractive. One of the most effective direct methods to construct solitary wave solutions of soliton systems is the tanh method firstly introduced by Li et al. [] and Malfliet. [3] The method was later extended by Ma [4] and Fan [5] and Lü et al. [6,7] In this paper, our concern is of the (+1)-dimensional Broer Kaup system with variable coefficients (VCBK), [8] u yt = α(t)(u xxy (u u x ) y v xx ), v t = α(t)( v xx (v u) x ), (1) where α(t) is an arbitrary nonzero function of time t. It is evident that when α(t) = 1, the above VCBK system becomes the well-known (+1)-dimensional Broer Kaup (BK) system, which may be derived from the inner-parameter-dependent symmetry constraint of the Kadomtsev Petviashvili model. [9] When α(t) = 1, system (1) becomes the (+1)-dimensional modified dispersive water-wave system. [10] When y = x, the (+1)- dimensional BK is reduced to the usual (1+1)-dimensional BK system, which is often used to describe the propagation of long waves in shallow water. [11] Using some suitable dependent and independent variable transformations, Chen and Li [1] have proved that the (+1)-dimensional BK system can be further transformed to the three dispersive long wave equation and three dimensional Ablowitz Kaup Newell Segur system. The (+1)-dimensional BK system has been widely investigated in details by many researchers. [13 17] However, to the best of our knowledge, the studies on the (+1)-dimensional VCBK system were reported in few papers. [8,18] In this paper, we will use the further extended tanh (FE-tanh) method, [6,7] which has been proven to be a powerful method for constructing explicit exact soliton-like solutions of high-dimensional nonlinear evolution equations, to construct soliton-like solutions, multi-soliton-like solutions, and some other types of solutions for the (+1)- dimensional VCBK system. Based on the obtained multisolutions, we will reveal some interesting multi-soliton structures. New Exact Solutions for (+1)-Dimensional VCBK System According to the FE-tanh method (see Refs. [6] and [7] for details), to get exact soliton-like solutions of system (1), we may take the transformation u = f + gφ(ω), v = F + Gφ(ω) + Hφ (ω), () where f = f(x, y, t), g = g(x, y, t), F = F (x, y, t), G = G(x, y, t), H = H(x, y, t), and ω = ω(x, y, t) are functions to be determined, and φ satisfies equation dφ dω = δ + φ (3) with δ an arbitrary real constant. Substituting Eqs. () and (3) into Eq. (1) and collecting coefficients of polynomials of φ, then eliminating each coefficient, we have 6αω x H(ω x + g) = 0, 6αω x (ω x H ω x w y g + ω y g ) = 0, gω yt δ + f yt + αff yx αgω yxx δ + 4αω x G x δ + αω xx Gδ αω y g xx δ αω yx g x δ + ω t g y δ The project supported by China Postdoctoral Science Foundation under Grant No and National Key Basic Research Project of China under Grant No. 004CB address: jmj01001@yahoo.com.cn

2 No. 5 New Soliton-like Solutions and Multi-soliton Structures for Broer Kaup System with Variable Coefficients 803 αω x g yx δ αg y ω xx δ + αω y gδf x + αfg y ω x δ αf yxx + αf xx + αω y fg x δ + 4α(ω x ) Hδ + g t ω y δ + αf y gω x δ + αfgω yx δ + αf y f x + αω y g ω x δ αω y g(ω x ) δ = 0, αgf yx + αf y g x + g yt 4αω x gω yx δ + αg xx + 4αω xx Hδ αg y (ω x ) δ + ω t gω y δ + 4αω y fgω x δ αg yxx αω y gω xx δ + αfg yx + αg ω yx δ + αg y f x + 4α(ω x ) Gδ 4αω y ω x g x δ + 4αω y gg x δ + 8αω x H x δ + 4αg y gω x δ = 0, αω x g yx αgω yxx + 4αω x G x αω y g xx + αgg yx + αg y g x + αf y gω x + g t ω y 8αω y g(ω x ) δ + αω y fg x + αh xx αg y ω xx + αω xx G + αfgω yx αω yx g x + αω y gf x + ω t g y + 8αω y g ω x δ + gω yt + αfg y ω x + 16α(ω x ) Hδ = 0, 4αω y ω x g x + 4α(ω x ) G + 4αω xx H αg y (ω x ) + 8αω x H x + ω t gω y αω y gω xx + 4αω y gg x 4αω x gω yx + αg ω yx + 4αg y gω x + 4αω y fgω x = 0, αω x fgδ + ω t Gδ + F t + αω x G x δ + αω xx Gδ + αω x gδf + α(ω x ) Hδ + αff x + αf x F + αf xx = 0, αg x F + αf x G + α(ω x ) Gδ + αfg x + αω xx Hδ + ω t Hδ + αgf x + G t + αg xx + 4αω x fhδ + 4αω x ggδ + 4αω x H x δ = 0, αfh x + αf x H + ω t G + αω x G x + H t + αω xx G + αω x gf + αgg x + αg x G + αh xx + αω x fg + 8α(ω x ) Hδ + 6αω x ghδ = 0, 4αω x gg + ω t H + 4αω x fh + 4αω x H x + αg x H + αω xx H + αgh x + α(ω x ) G = 0. (4) By the first equation of Eqs. (4) we get g = ω x. (5) Then we reduce Eqs. (4) by Eq. (5) to obtain and G = ω xy, H = ω x ω y, F = δω x ω y + f y, (6) ω t = αfω x αω xx, αδω yx ω xx αδω x ω yxx + αf y f x + αff yx + αf yxx + f yt = 0. (7) We distinguish different special cases to solve Eqs. (7). Case 1.1 f y = 0 and δ 0. In this case equations (7) can be solved to get the following special solutions of w and f: ω = Φ(x, t) + Ω(y), f = Φ t + αφ xx, (8) or ω = (Ψ(t)x + Γ(t))Θ(y) + Υ(y), f = Ψ tx + Γ t αψ, (9) where Φ, Ω, Ψ, Γ, Θ, and Υ are arbitrary functions of the corresponding arguments with Φ x 0 and Ψ 0. As equation (3) possesses the solutions δ tanh( δω), δ < 0, δ coth( δω), δ < 0, φ = δ tan( δω), δ > 0, δ cot( (10) δω), δ > 0, 1/ω, δ = 0, considering Eqs. (), (5), (6), and (8) (10), we obtain the following explicit exact solutions of the VCBK system (1). For δ < 0, soliton-like solutions u 1 = Φ t + αφ xx + δφ x tanh( δ(φ + Ω)), v 1 = δφ x Ω y + δφ x Ω y tanh( δ(φ + Ω)) ; (11) u = Φ t + αφ xx + δφ x coth( δ(φ + Ω)), v = δφ x Ω y + δφ x Ω y coth( δ(φ + Ω)) ; (1) u 3 = Ψ tx + Γ t αψ + δψθ tanh( δ((ψx + Γ)Θ + Υ)), v 3 = δψθ((ψx + Γ)Θ y + Υ y ) + δψθ y tanh( δ((ψx + Γ)Θ + Υ)) + δψθ((ψx + Γ)Θ y + Υ y ) tanh( δ((ψx + Γ)Θ + Υ)) ; (13) u 4 = Ψ tx + Γ t αψ + δψθ coth( δ((ψx + Γ)Θ + Υ)), v 4 = δψθ((ψx + Γ)Θ y + Υ y ) + δψθ y coth( δ((ψx + Γ)Θ + Υ)) + δψθ((ψx + Γ)Θ y + Υ y ) coth( δ((ψx + Γ)Θ + Υ)). (14)

3 804 JI Ming-Jun and LÜ Zhuo-Sheng Vol. 44 For δ > 0, formal periodic solutions u 5 = Φ t + αφ xx δφ x tan( δ(φ + Ω)), v 5 = δφ x Ω y δφ x Ω y tan( δ(φ + Ω)) ; (15) u 6 = Ψ tx + Γ t αψ δψθ tan( δ((ψx + Γ)Θ + Υ)), v 6 = δψθ((ψx + Γ)Θ y + Υ y ) δψθ y tan( δ((ψx + Γ)Θ + Υ)) δψθ((ψx + Γ)Θ y + Υ y ) tan( δ((ψx + Γ)Θ + Υ)). (16) As the function tan is periodic and Ω, Υ are arbitrary in Eqs. (15) and (16), we omit the cot type solutions here. Case 1. f y = 0 and δ = 0. In this case, the second equation of Eqs. (7) holds identically while the first equation possesses the following special solution of w and f: ω = c 0 + c 1 Φ(x, t) + c Ω(y) + c 3 Φ(x, t)ω(y), f = Φ t + α Φ xx, (17) in which c 0, c 1, c, and c 3 are arbitrary constants not vanishing, while ω, Φ, and Ω are arbitrary functions of the corresponding arguments with Φ x 0. Thus we get the following exact solution of the VCBK system (1), u = Φ t + αφ xx (c 1 + c 3 Ω)Φ x + c 0 + c 1 Φ + c Ω + c 3 ΦΩ, (c 3 c 0 c 1 c )Φ x Ω y v = (c 0 + c 1 Φ + c Ω + c 3 ΦΩ). (18) Reference [18] investigated the VCBK system (1) and obtained a solution (misprinted in Ref. [18]) being the same as Eq. (18) by using the variable separation approach. [19] Based on the solution, they revealed some interesting localized structures of system (1). Specially choosing α = c 0 = 1, equation (18) is the same as the solution (11) presented in Ref. [17]. Note that there is also a misprint in Ref. [17]. The solution (11b) there should be instead of u 1 = κ Φ + Φ δ tanh Φx + Ψ + κ G(x, y, t) = G(x, y, t) = (C c 1 c )p x q y (1 + c 1 p + c q + Cpq), (A c 1 c )p x q y (1 + c 1 p + c q + Cpq). Reference [17] considered the localized structures of field G (v in this paper) determined by the upper corrected expression. Case.1 f x = 0 and δ 0. In this subcase we get one special solution of Eq. (7) as follows: ω = Φ(y)x + Ψ(y) + κ αdt, f = κ Φ, (19) where Φ(y) and Ψ(y) are arbitrary functions of y with Φ(y) 0, and κ is an arbitrary constant. Then by Eqs. (), (5), (6), (10), and (19) we can obtain the following explicit exact solutions of Eq. (1). For δ < 0, soliton-like solutions )) αdt, v 1 = Φ(Φ y x + Ψ y )δ + κφ y Φ + Φ )) y δ tanh Φx + Ψ + κ αdt + Φ(Φ y x + Ψ y )δ tanh Φx + Ψ + κ αdt)), (0) u = κ )) Φ + Φ δ coth Φx + Ψ + κ αdt, v = Φ(Φ y x + Ψ y )δ + κφ y Φ + Φ )) y δ coth Φx + Ψ + κ αdt + Φ(Φ y x + Ψ y )δ coth Φx + Ψ + κ αdt)). (1) For δ > 0, formal periodic solutions u 1 = κ ( ( Φ Φ δ tan δ Φx + Ψ + κ )) αdt, v 1 = Φ(Φ y x + Ψ y )δ + κφ y Φ Φ ( ( )) y δ tan δ Φx + Ψ + κ αdt ( ( Φ(Φ y x + Ψ y )δ tan δ Φx + Ψ + κ αdt)). () Case. f x = 0 and δ = 0. In this subcase equation (7) possesses the special solution, n f = p(y) + q(t), ω = Ω 0 (y) + exp(φ i (y)x Φ i (y) i (y, t) + Ω i (y)), (3) i=1

4 No. 5 New Soliton-like Solutions and Multi-soliton Structures for Broer Kaup System with Variable Coefficients 805 in which p, Ω 0, Φ i, and Ω i (with 1 i n) are arbitrary functions of y, q is arbitrary function of t, n is arbitrary positive integer, i (y, t) = ((p + q) + Φ i (y))αdt with the integration constant being 0. Therefore, according to Eqs. (), (5), (6), (10), and (3), we obtain the following multi-soliton-like solution of Eq. (1): u = f + ω x ω, v = f y + ω xy ω ω xω y ω, (4) in which f and ω satisfies Eq. (3). Case 3 f = ω. In this case equations (7) possess the exact solutions, f = ω = x + Φ(y) αdt + κ with Φ(y) being arbitrary function of y, and κ being arbitrary constant, or Φ(y)x f = ω = Φ(y) αdt + Ψ(y) with Φ(y) and Ψ(y) being arbitrary functions of y. Thus we can get some other exact solutions of Eq. (1), which will not be listed here. One may obtain more explicit exact solutions of system (1) by searching for more exact solutions of Eq. (3) [0] or Eq. (7). Figure 3 shows the fission of a V-shaped multi-soliton to a web-shaped multi-soliton. The soliton fusion and fission phenomena have been observed in many fields of physics such as plasma physics, nuclear physics, hydrodynamics, and so on. [4] 3 Some Special Multi-soliton Structures of (+1)-Dimensional VCBK System In this section, we reveal by figures some interesting multi-soliton structures for the field v expressed by solution (4) with f and ω satisfying Eq. (3). Fig. 1 Plot of v determined by Eqs. (3) and (4) with α = 1, p = 1, q = 1, Ω 0 = 1, n =, Ω 1 = y, Ω = y, Φ 1 = 1, and Φ = 1 at time t = 3. Figure 1 shows a Y-shaped multi-soliton. The Y- shaped multi-soliton has also been found in some other nonlinear models such as the KP equation [1] and the (+1)-dimensional Burgers equation. [,3] Figure shows a multi-soliton structure which has the property of fusion. With time developing, the webshaped multi-soliton fuses to a Y-shaped multi-soliton. Fig. Plot of multi-soliton fusion for v determined by Eqs. (3) and (4) with α = 1, p = 0, q = 0, Ω 0 = 1, n = 3, Ω 1 = y 10, Ω = y 34, Ω 3 = y 90, Φ 1 = 1, Φ =, and Φ 3 = 3 at time (a) t = 0, (b) t = 5, (c) t = 11, (d) t = 16.

5 806 JI Ming-Jun and LÜ Zhuo-Sheng Vol. 44 Fig. 3 Plot of multi-soliton fission for v determined by Eqs. (3) and (4) with α = 1, p = 1, q = 1, Ω 0 = 1, n = 3, Ω 1 = y + 30, Ω = y 4, Ω 3 = 3y + 30, Φ 1 = 1, Φ =, and Φ 3 = 1 at time (a) t = 1, (b) t = 8, (c) t = 16, (d) t = 5. 4 Conclusion In summary, by means of the FE-tanh method, abundant new soliton-like solutions, multi-soliton-like solutions, and other exact solutions for the (+1)-dimensional VCBK system are obtained. Based on the derived exact multi-soliton solutions, some interesting multi-soliton structures are revealed. In particular, choosing the arbitrary functions in other solutions, new soliton structures may be found. The FE-tanh method is a powerful method, which can be used to obtain different types of exact solutions for lots of nonlinear evolution equations systematically. However, the main difficulty of using this method lies in the solving of coefficient equations (Eqs. (4) in this paper). The overcoming of this difficulty will make the FE-tanh method more efficient. This needs further research. References [1] S.Y. Lou, Phys. Rev. Lett. 80 (1998) 507. [] H.B. Li and K.L. Wang, J. Phys. A 3 (1990) [3] W. Malfliet, Am J. Phys. 60 (199) 650. [4] W.X. Ma, Int. J. Non-linear Mech. 31 (1996) 39. [5] E.G. Fan, Appl. Math. J. Chin. Univ. Ser. B 16 (001) 149. [6] Z.S. Lü and H.Q. Zhang, Phys. Lett. A 307 (003) 69. [7] Z.S. Lü and H.Q. Zhang, Chaos, Solitons & Fractals 17 (003) 669. [8] J.L. Zhang, Y.M. Wang, et al., J. Atom. Mol. Phys. 0 (003) 9. [9] S.Y. Lou and X.B. Hu, J. Math. Phys. 38 (1997) [10] C.L. Zheng, Commun. Theor. Phys. (Beijing, China) 40 (003) 5. [11] V.E. Zakharov and J. Li, Appl. Mech. Tech. Phys [1] C.L. Chen and Y.S. Li, Commun. Theor. Phys. (Beijing, China) 38 (1998) 190. [13] T.C. Xia and H.Q. Zhang, Chaos, Solitons & Fractals 16 (003) 167. [14] J.F. Zhang and P. Han, Acta. Phys. Sin. 51 (00) 705 (in Chinese). [15] S.Y. Lou, J. Phys. A: Math. Gen. 35 (00) [16] H.M. Li, Commun. Theor. Phys. (Beijing, China) 39 (003) 513. [17] C.L. Bai and H. Zhao, Chaos, Solitons & Fractals 3 (005) 777. [18] J.P. Fang, C.L. Zheng, and L.Q. Chen, Commun. Theor. Phys. (Beijing, China) 4 (004) 175. [19] X.Y. Tang, S.Y. Lou, et al., Phys. Rev. E 66 (00) [0] F.D. Xie, Ying Zhang, et al., Chaos, Solitons & Fractals 4 (005) 57. [1] J. Lin and H.M. Li, Z. Naturf. A 57 (00) 99. [] X.Y. Tang and S.Y. Lou, Chin. Phys. Lett. 0 (003) 335. [3] C.Z. Xu and J.F. Zhang, Acta Phys Sin. 53(08) (004) 407 (in Chinese). [4] M. Hisakado, Phys. Lett. A 7 (1997) 87.