MODULATIONAL INSTABILITY OF AN AXISYMMETRICAL STATE IN A 2-DIMENSIONAL KERR MEDIUM

We consider spatial evolution of small perturbations of an axisymmetri c state in a stationary two-dimensional medium with attractive cubic n onlinearity. For the unperturbed state, we find a one-parameter family of exact weakly localized solutions. The perturbation is expanded ove r angular harmonics, and its growth along the radial coordinate is the n considered. In contrast to the well known case of the one-dimensiona l modulational instability, the integral gain of the radially growing perturbations converges. It is calculated in the adiabatic approximati on, which is valid when the amplitude A of the unperturbed state and t he azimuthal ''quantum number'' of the perturbation are both large. In this approximation, the integral gain does not depend upon nz, and it increases linearly with A.