Instability of vortex solitons for 2D focusing NLS

We study instability of a vortex soliton $e^{i(mθ+ωt)}ϕ_{ω,m}(r)$ to $iu_t+Δu+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$ where $n=2$, $m\in\N$ and $(r,θ)$ are polar coordinates in $\R^2$. 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 $ϕ_{ω,m}$ is nonnegative. We investigate a limiting profile of $ϕ_{ω,m}$ as $m\to\infty$ and prove that for every $p>1$, there exists an $m_*\in \N$ such that for $m\ge m_*$, a vortex soliton $e^{i(mθ+ωt)}ϕ_{ω,m}(r)$ becomes unstable to the perturbations of the form $e^{i(m+j)θ}v(r)$ with $1\ll j\ll m$.

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