Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation

TL;DR

This paper proves the stability of a sum of N peakons in the Degasperis-Procesi equation, extending previous results on single peakon stability by localizing the proof and considering well-separated peakons traveling at different speeds.

Contribution

It localizes the proof of peakon stability and extends it to the sum of multiple well-separated peakons in the Degasperis-Procesi equation.

Findings

Stability of a single peakon is localized and clarified.

Stability of N peakons with large separation is established.

Results extend previous stability proofs to multi-peakon configurations.

Abstract

The Degasperis-Procesi equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Lin and Liu in [5]. In this paper, we localize the proof (in some suitable sense detailed in Section 3) of the stability of a single peakon. Thanks to this, we extend the result of stability to the sum of N peakons traveling to the right with respective speeds c1, . . . , cN , such that the difference between consecutive locations of peakons is large enough.