Whitham-broer-kaup Equation With Traveling Wave Solution Branch

In this paper, the bifurcations of solitary wave and kink waves for Whitham-Broer-Kaup equations is studied by using the bifurcation theory of planar dynamical systems. The existence of solitary waves and kink waves are proved. Explicit exact parametric representations of solitary waves, kink waves and periodic are obtained.This paper consists of four sections. In section 1 the paper introduce the corresponding model and results. In section 2 we discuss bifurcations of phase portraits (1.1.4). In section 3, twelve types explicit formulas of solitary wave solutions and kink wave solutionsof nine parametric conditions are given;In section 4, periodic solutions of (1.9) are given.