Journal of Mathematics of Kyoto University

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

Mizumachi, Tetsu

Full-text: Open access

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.

Article information

Source
J. Math. Kyoto Univ., Volume 48, Number 3 (2008), 471-497.

Dates
First available in Project Euclid: 14 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250271380

Digital Object Identifier
doi:10.1215/kjm/1250271380

Mathematical Reviews number (MathSciNet)
MR2511047

Zentralblatt MATH identifier
1175.35138

Citation

Mizumachi, Tetsu. Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 48 (2008), no. 3, 471--497. doi:10.1215/kjm/1250271380. https://projecteuclid.org/euclid.kjm/1250271380


Export citation