Orbital stability of smooth solitary waves for the $b$-family of Camassa-Holm equations

TL;DR

This paper analytically proves the orbital stability of smooth solitary waves in the $b$-family of Camassa-Holm equations for all $b>1$, addressing an open problem in the field.

Contribution

It provides a rigorous analytical verification of the stability criterion for smooth solitary waves in the $b$-Camassa-Holm equations, extending known results.

Findings

Smooth solitary waves are orbitally stable for all $b>1$

Analytical proof based on monotonicity of the period function

Addresses an open problem in the stability of these waves

Abstract

In this paper, we study the stability of smooth solitary waves for the $b$-family of Camassa-Holm equations. We verify the stability criterion analytically for the general case $b>1$ by the idea of the monotonicity of the period function for planar Hamiltonian systems and show that the smooth solitary waves are orbitally stable, which gives a positive answer to the open problem proposed by Lafortune and Pelinovsky [S. Lafortune, D. E. Pelinovsky, Stability of smooth solitary waves in the $b$-Camassa-Holm equation].