Stability of peakons for the generalized modified Camassa-Holm equation

TL;DR

This paper proves the orbital stability of peakons for the generalized modified Camassa-Holm equation, extending previous results to higher-order cases and introducing new methods for stability analysis.

Contribution

It introduces a novel approach to establish the stability of peakons for a higher-order generalization of the mCH equation, expanding understanding of nonlinear dispersive waves.

Findings

Single peakons are weak solutions of the PDEs.

The stability of peakons is verified using a new polynomial inequality approach.

Extension of stability results from mCH to higher-order gmCH equation.

Abstract

In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function $h(t,\,x)$ and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997, 2013) successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.