A note on traveling wave solutions to the two component Camassa-Holm equation

TL;DR

This paper investigates the relationship between traveling wave solutions of the two-component Camassa-Holm equation and the original Camassa-Holm equation, showing conditions under which solutions are shared and identifying new solutions.

Contribution

It establishes that non-smooth solutions are shared only under measure-zero conditions and introduces new smooth and distributional solutions for the two-component version.

Findings

No new peakon or cuspon solutions beyond Camassa-Holm

Shared solutions occur when the measure of $u^{-1}(c)$ is zero

New solutions exist when the measure of $u^{-1}(c)$ is non-zero

Abstract

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa-Holm equation are distributional traveling wave solutions to the Camassa-Holm equation provided that the set $u^{-1}(c)$, where $c$ is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa-Holm equation. However, the two component Camassa-Holm equation has distinct from Camassa-Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of $u^{-1}(c)$ is not zero. We provide examples of such solutions.