Asymptotic stability of solitons to 1D Nonlinear Schrodinger Equations in subcritical case

TL;DR

This paper proves the asymptotic stability of solitary waves in 1D nonlinear Schrödinger equations within the subcritical regime, employing advanced vector fields and Sobolev norm techniques to handle weak decay and nonlinearity.

Contribution

It introduces a novel combination of vector fields and Sobolev norm growth methods to establish stability in the challenging 1D subcritical setting.

Findings

Proves asymptotic stability of solitons in 1D NLS

Develops a method to handle weak decay in 1D

Controls high Sobolev norms growth over time

Abstract

In this paper, we prove the asymptotic stability of solitary waves to 1D nonlinear Schr\"odinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with respect to $t$ of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schr\"odinger equations obtained by Staffilani to control the high moments of the solutions emerging from the vector fields method.