Bifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component Systems

Abstract

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its [Formula: see text]-traveling wave system has a pseudo-peakon wave solution. In addition, its [Formula: see text]-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].