Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential

Abstract

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations \[ \begin{array}{cc} iu_{t} +\Delta u = V u \pm |u|^{p-1}u & \textrm{for } (x, t) \in \mathbb{R}^{2}\times \mathbb{R}, \end{array} \] in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [14] in the $n$-dimensional case ($n\geq 3$) by using the endpoint Strichartz estimate. Since the endpoint Strichartz estimate fails in 2-dimensional case, we use a time-global local smoothing estimate of Kato type to prove the asymptotic stability of a solitary wave.