Instability of vortex solitons for 2D focusing NLS

Abstract

We study instability of a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ to $$iu_t+\Delta u+|u|^{p-1}u=0,\quad\text{for $x\in \mathbb{R}^n$, $t>0$,}$$ where $n=2$, $m\in \mathbb{N}$ and $(r,\theta)$ are polar coordinates in $\mathbb{R}^2$. Grillakis \cite{Gr} proved that every radially symmetric standing wave solution is unstable if $p>1+4/n$. However, we do not have any examples of unstable standing wave solutions in the subcritical case $(p <1+n/4)$. Suppose $\phi_{\omega,m}$ is nonnegative. We investigate a limiting profile of $\phi_{\omega,m}$ as $m\to\infty$ and prove that, for every $p>1$, there exists an $m_*\in \mathbb{N}$ such that, for $m\ge m_*$, a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ becomes unstable under perturbations of the form $e^{i(m+j)\theta}v(r)$ with $1\ll j\ll m$.