Dynamics of collapse of optical pulses in Kerr medium

We consider the propagation of optical pulses in Kerr medium in the presence of a harmonic potential for 2D case in framework of Variational Approximation (VA). We will use two types of trial function: gaussian and hyperbolic secant ones. We will show that if the value of the nonlinearity parameter reaches a certain critical value the pulse collapses. This collapse corresponds to the physical situation when the self-focusing effect dominates over the dispersion and diffraction. We show that our analytical results are confirmed by several numerical calculations which are in excellent agreement with those of the VA predictions. By the analogy between the propagation equation and Gross-Pitajevski equation for Bose-Einstein condensates (BECs) we can transfer obtained results to the case of BEC placed in a external harmonic potential. It follows that in both of cases the wave collapse can appear.