Bifurcation on Smooth and Non-smooth Traveling Wave Solutions for a Generalized Two-component Camassa-Holm Equation

A generalized two-component Camassa-Holm equation is introduced as a model for shallow water waves moving over a linear shear flow. Bifurcations of traveling wave solutions are studied. Phase portraits of the traveling wave system are given. By using the method of planar dynamical systems, the existence of solitary wave solutions, smooth and non-smooth periodic traveling wave solutions is presented in different parametric conditions. Numerical simulations are made to agree the theoretical analysis. It shows that the existence of singular straight lines for the generalized two-component Camassa-Holm equation is the original cause of the non-smooth solutions. The existence of uncountably infinitely many breaking traveling wave solutions are given.