Two types of bounded traveling-wave solutions of a two-component Camassa–Holm equation

The Camassa–Holm equation (CH) is a well known integrable equation, with is used to describe the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of peakons from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in dynamics. In this paper, the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH2 equation. Two types of bounded traveling waves are found. One of them is called compacton, another is called generalized kink wave. The planar graphs of compactons and generalized kink awaves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink waves solutions are given.