Wave breaking and traveling waves for the quadratic-cubic Camassa–Holm equation

Abstract

This paper is concerned with the solutions of the quadratic-cubic Camassa–Holm equation which is a model that explore the change in the physical structure of the solutions from the peakons to the bell-shaped solitary wave solutions. The first type of solutions exhibits finite time singularity in the sense of wave breaking. We perform a refined analysis based on the local structure of the dynamics to provide a condition on the initial data to guarantee wave breaking. The key feature of the method is to refine the analysis on characteristics and conserved quantities to the Riccati-type differential inequality. The other type of solutions which we study is the traveling waves, we investigate nonexistence of the Camassa–Holm-type peaked traveling wave solutions. Moreover, we discover how the symmetric structure is connected to the steady structure of solutions to the quadratic-cubic Camassa–Holm equation, and prove that the classical symmetric waves must be traveling wave solutions.