Stability of peakons of the Camassa-Holm equation beyond wave breaking

TL;DR

This paper proves the global-in-time orbital stability of peakons in the Camassa-Holm equation, including after wave breaking, by analyzing the evolution of solutions and their energy measures.

Contribution

It extends previous H1-stability results to a global-in-time context, accounting for wave breaking and the stability of the energy measure's singular part.

Findings

Peakons are globally orbitally stable under perturbed solutions.

Stability persists even after wave breaking occurs.

The singular part of the energy measure remains stable over time.

Abstract

Using a generalized framework that consists of evolution of the solution to the Camassa- Holm equation and its energy measure, we establish the global-in-time orbital stability of peakons with respect to the perturbed (energy) conservative solutions to the Camassa-Holm equation. Especially, we extend the H1-stability result obtained by Constantin and Strauss (Comm. Pure Appl. Math., 53(5), 603-610, 2000) globally-in-time, even after the perturbed solutions experience wave breaking. In addition, our result also shows that the singular part of the energy measure of the perturbed solutions will remain stable for all times.