Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile

TL;DR

This paper extends the understanding of the asymptotic stability of peakons in the Camassa-Holm equation, especially for non-positive perturbations, and proves stability for antipeakon-peakon profiles with applications to energy decay.

Contribution

It broadens the class of initial data for which asymptotic stability of peakons is established, including non-positive perturbations and antipeakon-peakon trains.

Findings

Extended stability results to functions with non-positive momentum density parts.

Proved asymptotic stability of antipeakon-peakon profiles.

Showed energy decay on the left of any point for non-negative momentum density.

Abstract

We continue our investigation on the asymptotic stability of the peakon. In a first step we extend our asymptotic stability result [29] in the class of functions whose negative part of the momentum density is supported in ] -- $\infty$, x 0 ] and the positive part in [x 0 , +$\infty$[ for some x 0 $\in$ R. In a second step this enables us to prove the asymptotic stability of well-ordered train of antipeakons-peakons and, in particular, of the antipeakon-peakon profile. Finally, in the appendix we prove that in the case of a non negative momentum density the energy at the left of any given point decays to zero as time goes to +$\infty$,. This leads to an improvement of the asymptotic stability result stated in [29].