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

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.