November 19, 2013 – Gadi Fibich, Tel Aviv

Title: Continuations of the nonlinear Schrödinger equation beyond the singularity

The nonlinear Schrödinger equation (NLS) is the canonical model for propagation of intense laser beams. It has been known since 1965 that when the input power (L2 norm) is sufficiently high, NLS solutions blow up (collapse/become singular). While intense laser beams do undergo optical collapse, they continue to propagate beyond the NLS singularity point.  The standard approach for continuing NLS solutions beyond the singularity has been to consider more comprehensive models, in which the collapse is arrested. Motivated by the theory of vanishing-viscosity solutions in shock-wave theory, we recently adopted a different approach, and asked whether singular NLS solutions can be continued beyond the singularity, within the framework of the NLS model.  In this talk I will present some potential continuations of singular NLS solutions. A universal property of all continuations is that after the singularity, the solution loses its initial phase information. This phase-loss phenomenon implies that interactions between post-collapse beams become chaotic. Recent experimental confirmation of this theoretical prediction will be presented.  Joint work with M. Klein, B Shim, S.E. Schrauth, and A.L. Gaeta