The generalized peakon solution for the rotation-two-component Camassa–Holm system

Abstract

This paper is concerned with the rotation-two-component Camassa–Holm (R2CH) system, which is a model for equatorial water waves under the influence of the Coriolis force. The system includes the Dullin–Gottwald–Holm equation, the standard two-component integrable Camassa–Holm (CH) system and the CH equation as special cases. We aim to explore whether the R2CH system admits some generalized peakon weak solutions in the sense of distribution. The exact weak solutions in a particular form are derived by the ansatz method and they are proven to be the generalized peakon weak solutions. These solutions may help to explain the wave-breaking phenomenon in the wave motion to some extent. The method proposed in this paper is effective, which might be applied to study other nonlinear wave equations directly and even to be extend to search for fractal solitary wave.