Spatial instabilities in the propagation of a cylindrical beam in a Kerr medium

We study the development of spatial instabilities during the propagation of a cylindrically-symmetric beam in a medium with Kerr nonlinearity. Analytical results show that the initial evolution of the associated ring structure can be described by a Bessel function. Numerical simulations confirm these results and show that the rings increase very rapidly in sharpness and intensity. Their radii decrease as the beam propagates and the central ring collapses after a short distance.