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

Sondre Tesdal Galtung, Katrin Grunert

Published 2021-06-29Version 1

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 the traveling waves known as stumpons are not conservative. We illustrate this result by comparing the stumpon to simulations produced by a numerical scheme for conservative solutions, which has been recently developed by Galtung and Raynaud.