Orbital stability of periodic peakons for a new higher-order $\mu$-Camassa-Holm equation

TL;DR

This paper proves the orbital stability of periodic peakons in a new higher-order $$-Camassa-Holm equation, extending known properties of related equations and using conservation laws and inequalities.

Contribution

It introduces a higher-order $$-Camassa-Holm equation and demonstrates the orbital stability of its periodic peakons, a novel extension of existing models.

Findings

Periodic peakons are orbitally stable under small perturbations.

Stability is established using inequalities related to conservation laws.

The equation retains key properties of the original $$-Camassa-Holm models.

Abstract

Consideration here is a higher-order $\mu$-Camassa-Holm equation, which is a higher-order extension of the $\mu$-Camassa-Holm equation and retains some properties of the $\mu$-Camassa-Holm equation and the modified $\mu$-Camassa-Holm equation. By utilizing the inequalities with the maximum and minimum of the solution related to the first three conservation laws, we establish that the periodic peakons of this equation are orbitally stable under small perturbations in the energy space.