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Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation. (English) Zbl 1112.76010
Summary: We consider the Hirota-Satsuma equation for shallow water waves. We first obtain the Bäcklund transformation and Lax pairs by using the extended homogeneous balance method. Then we find some explicit exact solutions by means of Bäcklund transformation and the extended hyperbolic function method. These solutions include the solitary wave solution of rational function type, soliton solutions, double-soliton solutions, $N$-soliton solutions, the multiple solitary wave solutions, singular solutions, and periodic wave solutions of triangle function type.

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M60 Symmetry analysis, Lie group and algebra methods (fluid mechanics) 76B25 Solitary waves (inviscid fluids) 35Q51 Soliton-like equations
RAEEM
Full Text:
##### References:
 [1] Hirota, R.; Satsuma, J.: N-soliton solutions of model equations for shallow water waves. J. phys. Soc. jpn. 40, 611-614 (1976) [2] Hirota, R.; Satsuma, J.: Nonlinear equations generated from the Bäcklund transformation for the Boussinesq equation. Prog. theor. Phys. 57, 797-800 (1977) · Zbl 1098.81547 [3] Whitham, G. B.: Linear and nonlinear waves. (1974) · Zbl 0373.76001 [4] Clarkson, P. A.; Mansfield, E. L.: On a shallow water wave equation. Nonlinearity 7, 975-1000 (1994) · Zbl 0803.35111 [5] Satsuma, J.; Kaup, D. J.: Bäcklund transformation in bilinear form to the Hirota -- satsuma equation. J. phys. Soc. jpn. 43, 692 (1978) [6] Musette, M.; Conte, R.: Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations. J. math. Phys. 32, No. 6, 1450-1457 (1991) · Zbl 0734.35086 [7] Zhang, Yi; Deng, S. F.; Chen, Dengyuan: The novel multi-soliton solutions of equation for shallow water waves. J. phys. Soc. jpn. 72, No. 3, 763-764 (2003) · Zbl 1059.35109 [8] Zhang, Jinshun; Yang, Yunping: Darboux transformation of Hirota -- satsuma equation and its solutions. J. zhengzhou univ. (Sci.) 35, No. 3, 1-4 (2003) · Zbl 1059.35003 [9] Zhang, Yi; Chen, Dengyuan: Bäcklund transformation and soliton solutions for the shallow water wave equation. Chaos, solitons and fractals 20, 343-351 (2004) · Zbl 1046.35106 [10] Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering. (1991) · Zbl 0762.35001 [11] Wang, Mingliang: Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. lett. A 216, 67-74 (1996) [12] Wang, Mingliang: Exact solutions for a compound KdV -- Burgers equation. Phys. lett. A 213, 279-287 (1996) · Zbl 0972.35526 [13] Wang, Mingliang; Zhou, Yubin; Zhang, Huiqun: A nonlinear transformation of the shallow water wave equations and its application. Adv. math. (China) 28, No. 1, 72-75 (1999) · Zbl 1054.35518 [14] Fan, Engui; Zhang, Hongqin: A new approach to Bäcklund transformations of nonlinear evolution equations. Appl. math. Mech. 19, No. 7, 645-650 (1998) · Zbl 0923.35157 [15] Conte, R.; Musette, M.: Link between solitary waves and projective Riccati equations. J. phys. A: math. Gen. 25, 5609-5612 (1992) · Zbl 0782.35065 [16] Zhang, Guixu; Li, Zhibin; Duan, Yishi: Exact solitary wave solutions of nonlinear exact solitary wave solutions of nonlinear wave equations. Sci. China 44, No. 3, 396-401 (2001) · Zbl 1054.35032 [17] Yan, Zhenya: Generalized method and its application in the higher-order nonlinear Schrödinger equation in nonlinear optical fibres. Chaos, solitons and fractals 16, 759-766 (2003) · Zbl 1035.78006 [18] Chen, Yanze; Ding, Xinwei: Exact travelling wave solutions of nonlinear evolution equations in (1+1) and (2+1) dimensions. Nonlinear anal. 61, 1005-1013 (2005) · Zbl 1086.34501 [19] Yadong Shang, The extended hyperbolic functions method and exact solutions to the long -- short wave resonance equations, Chaos, Solitons and Fractals, in press, doi:10.1016/j.chaos.2006.07.007. · Zbl 1153.35374 [20] Li, Zhibin; Liu, Yinping: RAEEM: A Maple package for finding a series of exact travelling wave solutions for nonlinear evolution equations. Comput. phys. Commun. 163, 191-201 (2004) · Zbl 1196.35009