Asymptotic stability of solitary waves for the b-family of equations
Jun Wu, Yue Liu, Zhong Wang
TL;DR
The paper proves the asymptotic stability of lefton solitary waves for the nonintegrable b-family of one-dimensional fluid equations in the regime b<-1. It extends the Martel–Merle approach to a nonlocal, nonintegrable setting by establishing linear and nonlinear Liouville properties for perturbations around leftons, supported by detailed spectral analysis of the linearized operator and monotonicity arguments in weighted spaces. The authors then combine modulation theory with these rigidity results to show that small perturbations converge to a lefton (up to translation and amplitude scaling) in H^1, with exponential decay in certain spatial regions, marking the first rigorous asymptotic stability result for leftons in this nonintegrable regime. These results bridge nonlocal dispersive dynamics and stability theory for a broad class of solitary waves in fluid models, highlighting the delicate interplay between nonlocality, spectral structure, and nonlinear stability.
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.