Asymptotic Stability and Completeness in the Energy Space for Nonlinear Schrödinger Equations with Small Solitary Waves

2003/08/06 by Stephen J. Gustafson, Stephen Gustafson, Kenji Nakanishi +5
Mathematics · Physics and Astronomy · #35Q40 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #msc:35Q40 #msc:35Q55

paper · pdf · doi:10.48550/arxiv.math-ph/0308009

arxiv created 2003/08/06 · openalex publication_date 2003/08/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study a class of nonlinear Schrödinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H¹, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as t → ∞, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges to a solution of the free Schrödinger equation.

Citations

• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions

Related

• On Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation
• Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation
• Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential
• Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes
• Solitary waves for the power degenerate NLS -- existence and stability