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 $p>3$.