Asymptotic stability of solitary waves for the b-family of equations

TL;DR

This paper proves the asymptotic stability of lefton solitary waves in the non-integrable $b$-family of equations for $b < -1$, using adapted analytical techniques from generalized KdV equations.

Contribution

It provides the first rigorous proof of asymptotic stability for lefton solutions in the non-integrable $b$-family of equations, extending stability theory beyond integrable cases.

Findings

Lefton solutions are asymptotically stable for $b < -1$.

The analysis adapts Martel-Merle framework to nonlocal, non-integrable equations.

First stability results for leftons outside integrable regimes.

Abstract

We establish the asymptotic stability of lefton solutions-exponentially localized stationary solitary waves-for the $b$-family of equations with positive momentum density in the regime $b < -1$. Unlike the completely integrable Camassa-Holm $(b=2)$ and Degasperis-Procesi $(b=3)$ cases, this parameter range lies outside integrability and exhibits distinct nonlinear dynamics. Our analysis adapts the Martel-Merle framework for generalized KdV equations to the nonlocal, non-integrable structure of the $b$-family of equations. The proof combines a nonlinear Liouville property for solutions localized near leftons with a refined spectral analysis of the associated linearized operator. These results provide the first rigorous asymptotic stability theory for leftons in the non-integrable $b$-family of equations.