Asymptotic stability of solitons for near-cubic NLS equation with an internal mode

2024/04/22 by Guillaume Rialland, Rialland, Guillaume · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2404.13980

openalex publication_date 2024/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider perturbations of the one-dimensional cubic Schrödinger equation, of the form i ∂_t ψ+ ∂_x² ψ+ |ψ|² ψ+ g( |ψ|² ) ψ= 0. Under hypotheses on the function g that can be easily verified in some cases (such as g(s) = s^σ with σ>1), we show that the linearized problem around a small solitary wave presents a unique internal mode. Moreover, under an additional hypothesis (the Fermi golden rule) that can also be verified in the case of powers g(s) = s^σ, we prove the asymptotic stability of the solitary waves with small frequencies.

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