New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water

Based upon the well-known Riccati equation, a new generalized transformation is presented and applied to solve Whitham-Broer-Kaup (WBK) equation in shallow water As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained. And variant Boussinesq equation and the system of approximate equation for long water waves, as the special cases of WBK equation, can also obtain the corresponding solitary wave solutions and periodic wave solutions. In addition, with the aid of Mathematica and Wu elimination method to solve a large system of algebraic equations, the course of solving equations can be carried out in computer. (C) 2001 Published by Elsevier Science B.V.