Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension

Abstract

A perturbation analysis with respect to the space dimension is used to construct singular solutions of the two-dimensional Schr\"odinger equation with cubic nonlinearity. These solutions blow up at a rate {ln ln[(${\mathit{t}}^{\mathrm{*}}$-t${)}^{\mathrm{\ensuremath{-}}1}$]/(${\mathit{t}}^{\mathrm{*}}$-t)${\mathrm{}}}^{1/2}$, in contrast to the behavior in three dimensions where there is no logarithmic correction. The form of such solutions is supported by the results of high-resolution numerical simulations.