Vortex solitons for 2D focusing nonlinear Schrödinger equation. (Q976605)

Standing wave solutions of the form \(\text{e}^{\text{i}(\omega t+m\theta)}\phi _\omega (r)\) to the nonlinear Schrödinger equation \[ \text{i}u_t+\Delta u+| u| ^{p-1}u=0 \] for \(x\in \mathbb R^2\) and \(t>0\) are studied. Here \((r,\theta)\) are polar coordinates and \(m\in \mathbb N\cup \{0\}\). It is proved that standing waves which have no node are unique for each \(m\) and that they are unstable if \(p>3\).