Stability of Solitary Waves and Wave-Breaking Phenomena for the Two-Component Camassa–Holm System

Considered herein is a two-component Camassa–Holm system modeling shallow water waves moving over a linear shear flow. It is shown here that solitary-wave solutions of the system are dynamically stable to perturbations for a range of their speeds. On the other hand, a new wave-breaking criterion for solutions is established, and two results of wave-breaking solutions with certain initial profiles are described in detail. Moreover, a sufficient condition for global solutions determined only by a nonzero initial profile of the free surface component of the system is found.