Continuations of the nonlinear Schrödinger equation beyond the singularity

We present four continuations of the critical nonlinear Schrödinger equation (NLS) beyond the singularity: (1) a sub-threshold power continuation, (2) a shrinking-hole continuation for ring-type solutions, (3) a vanishing nonlinear-damping continuation and (4) a complex Ginzburg–Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that lead to a loglog collapse, the sub-threshold power limit is a Bourgain–Wang solution, both before and after the singularity, and the vanishing nonlinear-damping and CGL limits are a loglog solution before the singularity, and have an infinite-velocity expanding core after the singularity. Our results suggest that all NLS continuations share the universal feature that after the singularity time Tc, the phase of the singular core is only determined up to multiplication by eiθ . As a result, interactions between post-collapse beams (filaments) become chaotic. We also show that when the continuation model leads to a point singularity and preserves the NLS invariance under the transformation t → −t and ψ → ψ∗, the singular core of the weak solution is symmetric with respect to Tc. Therefore, the sub-threshold power and the shrinking-hole continuations are symmetric with respect to Tc, but continuations which are based on perturbations of the NLS equation are generically asymmetric. Mathematics Subject Classification: 35Q55 (Some figures in this article are in colour only in the electronic version) 0951-7715/11/072003+43$33.00 © 2011 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 2003 2004 G Fibich and M Klein