Asymptotic stability of small solitons to 1D NLS with potential

TL;DR

This paper proves the asymptotic stability of small solitary waves in 1D nonlinear Schrödinger equations with potential by combining local smoothing effects with Strichartz estimates.

Contribution

It introduces a novel approach using global local smoothing effects to establish stability, extending previous 3D results to 1D cases.

Findings

Proves global in time local smoothing effect for 1D NLS with potential.

Shows the dispersive part of solutions belongs to a specific function space.

Establishes asymptotic stability of small solitary waves in the energy class.

Abstract

We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schr\"{o}dinger equations $$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that the local smoothing effect of Kato type holds global in time and combine this estimate with the Strichartz estimate to show $\|(1+x^2)^{-3/4}v\|_{L^\infty_xL^2_t}<\infty$, which implies the asymptotic stability of a solitary wave.