Stumpons are non-conservative traveling waves of the Camassa-Holm equation

Abstract

It is well-known that by requiring solutions of the Camassa--Holm equation to satisfy a particular local conservation law for the energy in the weak sense, one obtains what is known as conservative solutions. As conservative solutions preserve energy, one might be inclined to think that any solitary traveling wave is conservative. However, in this paper we prove that this is not true for the traveling waves known as stumpons. We illustrate this result by comparing the stumpon to simulations produced by a recently developed numerical scheme for conservative solutions.