Asymptotic stability of solitary waves for the 1D focusing cubic Schr\"odinger equation

TL;DR

This paper proves the full asymptotic stability of solitary waves in the 1D focusing cubic Schrödinger equation, using advanced analytical techniques to handle slow decay and resonances.

Contribution

It introduces a novel combination of space-time resonances, distorted Fourier transform, and modulation methods to establish stability and modified scattering.

Findings

Proved asymptotic stability of solitary waves.

Demonstrated modified scattering for radiation.

Developed a moving-center local smoothing estimate.

Abstract

We establish the full asymptotic stability of solitary wave solutions for the 1D focusing cubic Schr\"odinger equation on the line under small perturbations in weighted Sobolev spaces, building upon our results in [58]. The proof integrates the space-time resonances approach, based on the distorted Fourier transform, with modulation techniques to show modified scattering for the radiation term and convergence for the modulation parameters. A key challenge throughout the nonlinear analysis is the slow local decay of the radiation term, caused by threshold resonances in the linearized operator. The presence of favorable null structures in the quadratic nonlinearities mitigates this problem through the use of normal form transformations. Another essential step in the proof involves developing a variant of the local smoothing estimate that incorporates a moving center.