Integrable system modelling shallow water waves: Kaup–Boussinesq shallow water system

Extended F-expansion method is proposed to seek exact solutions of the integrable nonlinear Kaup–Boussinesq shallow water system, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the extended F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously a number of periodic wave solutions expressed by various Jacobi elliptic functions for the integrable nonlinear Kaup-Boussinesq shallow water system. In the limiting cases, the solitary wave solutions are obtained as well.