Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation

In this paper 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, soliton solutions, double-soliton solutions, N-soliton solutions, the multiple solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type.