## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 4 (2005), 431-450.

### Vortex solitons for 2D focusing nonlinear Schrödinger equation

#### Abstract

We study standing wave solutions of the form $e^{i(\omega t+m\theta)}\phi_\omega(r)$ to the nonlinear Schrödinger equation $$iu_t+\Delta u+|u|^{p-1}u=0\quad\text{for $x\in \mathbb{R}^2$ and $t>0$,}$$ where $(r,\theta)$ are polar coordinates and $m\in\mathbb N\cup\{0\}$. We prove that standing waves which have no node are unique for each $m$ and that they are unstable if $p>3$.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 4 (2005), 431-450.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060196

**Mathematical Reviews number (MathSciNet)**

MR2122708

**Zentralblatt MATH identifier**

1212.35455

**Subjects**

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

Secondary: 35B35: Stability 35J60: Nonlinear elliptic equations 35Q51: Soliton-like equations [See also 37K40]

#### Citation

Mizumachi, Tetsu. Vortex solitons for 2D focusing nonlinear Schrödinger equation. Differential Integral Equations 18 (2005), no. 4, 431--450. https://projecteuclid.org/euclid.die/1356060196