On universality of blow-up profile for L2 critical nonlinear Schrodinger equation

We consider finite time blow-up solutions to the critical nonlinear Schrodinger equation iu(t) = -Deltau - |u|(4/N)u with initial condition u(0) is an element of H-1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23]. We establish in this paper the existence of a universal blow-up profile which attracts blow-up solutions in the vicinity of blow-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of (NLS) in L-2 will remove the possibility of self similar blow-up in energy space H-1.