New singular solutions of the nonlinear Schrödinger equation

Gadi Fibich, Nir Gavish, Xiao Ping Wang

Research output: Contribution to journalArticlepeer-review

Abstract

We present numerical simulations of a new type of singular solutions of the critical nonlinear Schrödinger equation (NLS), that collapse with a quasi self-similar ring profile at a square root blowup rate. We find and analyze the equation of the ring profile. 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. Collapsing ring solutions are also observed in the supercritical NLS.

Original languageEnglish
Pages (from-to)193-220
Number of pages28
JournalPhysica D: Nonlinear Phenomena
Volume211
Issue number3-4
DOIs
StatePublished - 15 Nov 2005

Keywords

  • Blowup rate
  • Collapse
  • Nonlinear Schrödinger equation
  • Ring profile
  • Self-similar solution
  • Singularity

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