GADI FIBICH
Abstract
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The study of singular solutions of the critical nonlinear
Schr\"odinger equation~(NLS) during the last twenty years
has lead to the belief that the only stable singular
solutions are those that collapse according to the loglog law
with a self-similar profile which is the
ground state of the equation $\Delta R -R + R^{4/d+1} = 0$.
In this study we present numerical simulations of a new type of singular
solutions of the NLS, that collapse with a self-similar ring profile
(which is different from the R profile) at a square
root blowup rate. We observe that the self-similar ring profile is an
attractor for a large class of radially-symmetric initial
conditions, but is unstable under symmetry-breaking perturbations.
The equation for the ring profile admits also multi-ring solutions
that give rise to collapsing self-similar multi-ring solutions,
but these solutions are unstable even in the radially-symmetric
case, and eventually collapse with a single ring profile.