Transient behavior of collapsing ring solutions in the critical nonlinear Schrödinger equation

The critical nonlinear Schrödinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.