## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 47, Number 3 (2007), 599-620.

### 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.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 47, Number 3 (2007), 599-620.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281026

**Digital Object Identifier**

doi:10.1215/kjm/1250281026

**Mathematical Reviews number (MathSciNet)**

MR2402517

**Zentralblatt MATH identifier**

1146.35085

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35B35: Stability 35Q51: Soliton-like equations [See also 37K40]

#### Citation

Mizumachi, Tetsu. Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 47 (2007), no. 3, 599--620. doi:10.1215/kjm/1250281026. https://projecteuclid.org/euclid.kjm/1250281026