Modulational instability of an axisymmetric state in a two-dimensional Kerr medium

We consider spatial evolution of small perturbations of an axisymmetric state in a stationary two-dimensional medium with attractive cubic nonlinearity. For the unperturbed state, we find a one-parameter family of exact weakly localized solutions. The perturbation is expanded over angular harmonics, and its growth along the radial coordinate is then considered. In contrast to the well known case of the one-diensional modulational instability, the integral gain of the radially growing perturbations converges. It is calculated in the adiabatic approximation, which is valid when the amplitude A of the unperturbed state and the azimuthal "quantum number" of the perturbation are both large. In this approximation, the integral gain does not depend upon m, and it increases linearly with A.