Revisiting asymptotic stability of solitons of nonlinear Schr\"odinger equations via refined profile method

TL;DR

This paper presents a new proof for the asymptotic stability of solitons in nonlinear Schrödinger equations with internal modes, utilizing refined profiles to avoid normal forms and track key functions in stability analysis.

Contribution

Introduces a novel proof method using refined profiles that bypasses normal forms and enables tracking of functions in the Fermi Golden Rule for soliton stability.

Findings

Proof avoids normal forms in stability analysis

Tracks functions in Fermi Golden Rule hypothesis

Provides an alternative approach to soliton stability

Abstract

In this paper, we give an alternative proof for the asymptotic stability of solitons for nonlinear Schr\"odinger equations with internal modes. The novel idea is to use "refined profiles" developed by the authors for the analysis of small bound states. By this new strategy, we able to avoid the normal forms. Further, we can track the functions appearing in the Fermi Golden Rule hypothesis.