On the formation of singularities in solutions of the critical nonlinear Schrodinger equation

For the one-dimensional nonlinear Schrodinger 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 in finite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form lambda (t) similar to (ln / ln(T* - t) /)(1/2) (T* - t)(-1/2).