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arXiv · 2512.01225

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

Jun Wu·Yue Liu·Zhong Wang
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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.

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Jun Wu, Yue Liu, Zhong Wang. 2025-12-01. Asymptotic stability of solitary waves for the b-family of equations. https://arxiv.org/abs/2512.01225

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