Stability of isotropic singularities for the nonlinear Schrödinger equation

Abstract

We describe a method by which the fully two- and three-dimensional cubic Schrödinger equations can be accurately integrated numerically up to times very close to the formation of singularities. In both cases, anisotropic initial data collapse very rapidly towards isotropic singularities. In three dimensions, the solutions become self-similar with a blowup rate ( t ∗−t) -1 2 . In two dimensions, the self-similarity is weakly broken and the blowup rate is [( t ∗−t)/ ln ln 1/(t ∗−t)] -1 2 . The stability of the singular isotropic solutions is very firmly backed by the numerical results.