## 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 Lie algebra methods applied to problems in fluid mechanics 76B25 Solitary waves for incompressible inviscid fluids 35Q51 Soliton equations

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### References:

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