Continuations of the nonlinear Schrödinger equation beyond the singularity

Details Continuations of the nonlinear Schrödinger equation beyond the singularity Continuations of the nonlinear Schrödinger equation beyond the singularity

2011-03-10

arxiv.org/abs/1103.1962v2

Mathematics

Analysis of PDEs

Scientific paper

We present four continuations of the critical nonlinear \schro 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 leads 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{\rev{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 $e^{i\theta}$. 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 $t\rightarrow-t$ and $\psi\rightarrow\psi^\ast$, the singular core of the weak solution is symmetric with respect to $T_c$. Therefore, the sub-threshold power and the{\rev{shrinking}}-hole continuations are symmetric with respect to $T_c$, but continuations which are based on perturbations of the NLS equation are generically asymmetric.

Affiliated with

Also associated with

No associations

Rating

Continuations of the nonlinear Schrödinger equation beyond the singularity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Continuations of the nonlinear Schrödinger equation beyond the singularity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continuations of the nonlinear Schrödinger equation beyond the singularity will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-630543