Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential

Abstract

We consider asymptotic stability of a small solitary wave to supercritical $1$-dimensional nonlinear Schrödinger 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{18} 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 a local smoothing estimate of Kato type holds globally in time and combine the estimate with the Strichartz estimate to show $\|(1+x^2)^{-3/4} v \|_{L^{\infty}_x L^2_t} < \infty$, which implies the asymptotic stability of a solitary wave.