Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called u-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the u-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.