On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation

Abstract.

For the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity the Cauchy problem with initial data close to a soliton is considered. It is shown that for a certain class of initial perturbations the solution develops a self-similar singularity infinite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form¶¶\( \lambda(t) \sim (ln \mid ln(T^* -t)\mid)^{1/2} (T^* - t)^{-1/2} $.