On an integrable two-component Camassa-Holm shallow water system

TL;DR

This paper introduces a two-component integrable Camassa-Holm system derived from shallow water theory, analyzing its solutions, wave breaking phenomena, solitary waves, and explicit peakon solutions in the short wave limit.

Contribution

It presents a new integrable two-component system, explores its solution behaviors, wave breaking, and constructs explicit peakon solutions, expanding understanding of multi-component shallow water models.

Findings

Small initial data lead to global solutions.

Certain initial data cause wave breaking.

Explicit peakon solutions are constructed in the short wave limit.

Abstract

The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this letter we deal with such a two-component integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.