Variable separation solution and soliton excitations of the (1+1)-dimensional generalised shallow water wave equation

In this paper, variable separation solution and soliton excitations of the (1+1)-dimensional generalised shallow water wave equation are obtained. This equation includes two special cases which are completely integrable (IST integrable): the AKNS equation and the Hirota-Satsuma equation. Firstly, the variable separation (BT-VS) method based on the Bcklund transformation is extended to this eqaution for deriving VS solutions which include some low dimensional arbitrary functions. In the integrable cases, a space arbitrary function and a time arbitrary function are included. But in the other cases only a time arbitrary function is included and the space function needs to satisfy a specific condition. In addition, for the (1+1)-dimensional universal formula, abundant soliton excitations can be constructed, such as one-soliton, bell-anti-bell soltion, soliton expansion, breather-like, instaton-like. Finally, some discusions are made about the VT-VS method.