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

TL;DR

This paper demonstrates that stumpons, a type of traveling wave in the Camassa-Holm equation, are non-conservative despite the common association of solitary waves with conservation, supported by numerical comparisons.

Contribution

It proves that stumpons are non-conservative traveling waves, challenging assumptions about solitary waves in the Camassa-Holm equation.

Findings

Stumpons are non-conservative traveling waves.

Numerical simulations confirm the non-conservative nature of stumpons.

Comparison with conservative solutions highlights differences in energy preservation.

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.