Breaking Waves And Solitary Waves To The Rotation-Two-Component Camassa--Holm System

Abstract

In this paper, we consider two types of solutions of the rotation-two-component Camassa--Holm (R2CH) system, a model of equatorial water waves that includes the effect of the Coriolis force. 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 some criteria that leads to the blow-up of solutions. The other type of solutions we study is the solitary waves. We classify various localized solitary wave solutions for the R2CH system. In addition to those smooth solitary wave solutions, we show that there are solitary waves with singularities, like peakons and cuspons, depending on the values of the rotating parameter $\Omega$ and the balance index $\sigma$. We also prove that horizontally symmetric weak solutions of this model must be traveling waves.