A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon

TL;DR

This paper establishes the asymptotic stability of peakons in the Degasperis-Procesi equation by proving a rigidity result for solutions shared by the Holm-Staley b-family, extending prior work on the Camassa-Holm equation.

Contribution

It introduces a new rigidity result applicable to the Holm-Staley b-family, demonstrating the stability of peakons in the Degasperis-Procesi equation.

Findings

Peakons are asymptotically H1-stable under the Degasperis-Procesi flow.

Rigidity results hold for solutions with exponential decay in the Holm-Staley b-family.

Extension of stability results from the Camassa-Holm to the Degasperis-Procesi equation.

Abstract

We prove that the peakons are asymptotically H 1-stable, under the flow of the Degasperis-Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm-Staley b-family of equations for b $\ge$ 1. This extends previous results obtained for the Camassa-Holm equation (b = 2).