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.Keywords: Camassa-Holm equation; traveling waves; peakons. Mathematics Subject Classification 2000: 35Q35, 35Q53The Camassa-Holm equation [1] arises as a model for the unidirectional propagation of shallow water waves over a flat bottom, u(x, t) representing the water's free surface, and κ ∈ R being a parameter related to the critical shallow water speed (see [13] and [12] for the derivation of this equation). Camassa and Holm [1] discovered that the equation has non-smooth solitary waves that retain their individual characteristics through the interaction and eventually emerge with their original shapes and speeds. The traveling wave solutions of the Camassa-Holm equation have been classified by Lenells [14]. The peaked traveling waves appear to have the characteristic of the traveling waves of greatest height similar to exact traveling wave solutions of the governing equations for water waves with a peak at their crest (see [17,3,4]). However, simpler approximate shallow water models like KdV do not admit traveling wave solutions with this feature (see [18]). Also, it should be noted that the Camassa-Holm peaked traveling waves are stable wave forms (see [5,6,15]). Therefore, they are physically recognizable. An alternative, and useful for generalizations form of this equation iswhere m = u − u xx + 1 2 κ.117