Non-self-similar collapsing solutions of the nonlinear Schrödinger equation at the critical dimension

The dynamical problem of a spherically symmetric wave collapse is investigated in the framework of the nonlinear Schr\"odinger equation defined at the critical dimension. Collapsing solutions are shown to remain self-similar for spatial coordinates below a cutoff radius only, and to exhibit at larger distances a non-self-similar tail whose expression is explicitly computed. A rapid method used to study the time behavior and the stability of the contraction rate associated with these singular solutions is also derived.