On a singular wave collapse

Spontaneous nonlinear wave field singularities in many cases do not vanish immediately after absorption of waves that formed them but exist for a longer time and suck in the new waves. The energy absorbed in this way may exceed considerably that expended in the formation of the singularities. In particular, the former energy may be finite under conditions when an infinitesimal amount of energy is involved in the formation of the singularities, i.e. when the collapse is weak. A more careful verification of these recent proposals is carried out by investigating the wave field dynamics in the space-time vicinity of the point of formation of the singularity. This is done within the framework of the nonlinear Schroedinger equation encountered in a number of physical problems. It is shown that in a broad range of parameters of the model the dynamics of the wave field in the aforementioned region is of a self-similar nature