A perturbation analysis with respect to the space dimension is used to construct singular solutions of the two-dimensional Schrödinger equation with cubic nonlinearity. These solutions blow up at a rate {ln ln[(${\mathit{t}}^{\mathrm{*}}$-t${)}^{\mathrm{-}1}$]/(${\mathit{t}}^{\mathrm{*}}$-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.

• Received 7 March 1988

DOI:https://doi.org/10.1103/PhysRevA.38.3837