Stability of smooth solitary waves in the b-Camassa-Holm equation

TL;DR

This paper establishes the stability conditions for smooth solitary waves in the b-Camassa-Holm equation, confirming stability analytically for integrable cases and numerically for non-integrable cases.

Contribution

It provides a precise stability criterion for smooth solitary waves in the b-family of Camassa-Holm equations, including both integrable and non-integrable cases.

Findings

Analytical proof of stability for b=2 and b=3.

Numerical and asymptotic evidence of stability for all b > 1.

Different Hamiltonian formulations used for stability analysis.

Abstract

We derive the precise stability criterion for smooth solitary waves in the b-family of Camassa-Holm equations. The smooth solitary waves exist on the constant background. In the integrable cases b = 2 and b = 3, we show analytically that the stability criterion is satisfied and smooth solitary waves are orbitally stable with respect to perturbations in $H^3(\mathbb{R})$. In the non-integrable cases, we show numerically and asymptotically that the stability criterion is satisfied for every b > 1. The orbital stability theory relies on a different Hamiltonian formulation compared to the Hamiltonian formulations available in the integrable cases.