Continuations of the nonlinear Schrödinger equation beyond the singularity

ISSN： 0951-7715

Source： Nonlinearity, Vol.24, Iss.7, 2011-07, pp. : 2003-2045

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Abstract

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 T_c, the phase of the singular core is only determined up to multiplication by e^iθ. 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 T_c. Therefore, the sub-threshold power and the shrinking-hole continuations are symmetric with respect to T_c, but continuations which are based on perturbations of the NLS equation are generically asymmetric.