Stability of smooth periodic traveling waves in the Camassa-Holm equation

TL;DR

This paper analyzes the stability of smooth periodic traveling waves in the Camassa-Holm equation, showing that a nonstandard Hamiltonian formulation provides clearer stability criteria and numerical verification confirms these conditions.

Contribution

It introduces a nonstandard Hamiltonian framework for analyzing periodic waves in the Camassa-Holm equation, improving stability analysis over the standard formulation.

Findings

The period function is monotone in the nonstandard formulation.

The quadratic energy form has only one negative eigenvalue.

Numerical results confirm the stability condition in the wave existence region.

Abstract

Smooth periodic travelling waves in the Camassa--Holm (CH) equation are revisited. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg--de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.