Instability of vortex solitons for 2D focusing NLS

Abstract

We study instability of a vortex soliton $e^{i(m\theta +\omega t)}{\varphi }_{\omega ,m}(r)$ to $$i{u}_t+\Delta u+|u{|}^{p-1}u=0,\text{for x∈Rn, t>0,}$$ where $n=2$, $m\in \mathbb{N}$ and $(r,\theta )$ are polar coordinates in ${\mathbb{R}}^2$. Grillakis \cite{Gr} proved that every radially standing wave solutions are 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 ${\varphi }_{\omega ,m}$ is nonnegative. We investigate a limiting profile of ${\varphi }_{\omega ,m}$ as $m\to \infty$ and prove that for every $p>1$, there exists an $m_{\ast }\in \mathbb{N}$ such that for $m\ge m_{\ast }$, a vortex soliton $e^{i(m\theta +\omega t)}{\varphi }_{\omega ,m}(r)$ becomes unstable to the perturbations of the form $e^{i(m+j)\theta }v(r)$ with $1\ll j\ll m$.

20pages, no figure