Stability of peakons for the Degasperis-Procesi equation

TL;DR

This paper proves the orbital stability of peakon solitons in the Degasperis-Procesi equation, demonstrating their robustness under small disturbances using a Lyapunov function approach.

Contribution

It establishes the orbital stability of peakons for the Degasperis-Procesi equation, a result not previously proven.

Findings

Peakons are orbitally stable under small perturbations.

Construction of a Lyapunov function confirms stability.

Results contribute to understanding soliton dynamics in shallow water models.

Abstract

The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Liapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.