Orbital stability of peakons for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

TL;DR

This paper proves the orbital stability of peakon solutions for a generalized Camassa-Holm equation with quadratic and cubic nonlinearities, demonstrating that solutions close to peakons remain close over time.

Contribution

It introduces a new stability analysis for peakons in a modified Camassa-Holm equation with combined nonlinearities, using polynomial inequalities and conserved quantities.

Findings

Peakons are orbitally stable under small perturbations.

A polynomial inequality relates conserved quantities and solution maxima.

Solutions starting near peakons stay close to some translate of the peakon.

Abstract

In this paper, we investigate the orbital stability problem of peakons for a modified Camassa-Holm equation with both quadratic and cubic nonlinearity. This equation was derived from integrable theory and admits peaked soliton (peakon) and multipeakon solutions. By constructing two suitable piecewise functions, we establish the polynomial inequality relating to two conserved quantities and the maximum of the solution to this equation. The error estimate between the maximum of the solution and the peakon then follows from the structure of the polynomial inequality. Finally, we prove that a wave starting close to the peakon remains close to some translate of it at all later times, that is, the shapes of these peakons are stable under small perturbations.