Orbital stability of peakon solutions for a generalized higher-order Camassa-Holm equation

TL;DR

This paper proves the orbital stability of peakon solutions for a generalized higher-order Camassa-Holm equation, extending understanding of nonlinear wave stability in higher-order dispersive equations.

Contribution

It demonstrates the existence and orbital stability of peakon solutions for a higher-order extension of the Camassa-Holm equation, using conserved quantities and non-sgn-changing conditions.

Findings

Existence of global weak peakon solutions.

Orbital stability in the energy space.

Enhanced understanding of higher-order nonlinear effects.

Abstract

In this paper, we investigate the orbital stability issue of a generalized higher-order Camassa-Holm (HOCH) equation, which is an higher-order extension of the quadratic CH equation. Firstly, we show that the HOCH equation admits a global weak peakon solution by paring it with ssome smooth test function. Secondly, with the help of two conserved quantities and the non-sgn-changing condition, we prove the orbital stability of this peakon solution in the energy space in the sense that its shape remains approximately the same for all times. Our results enrich the research of the orbital stability for the CH-type equations and are useful to better understand the impact of higher-order nonlinearities on the dispersion dynamics.