Ring-shaped solitons in a “dartboard” photonic lattice

We introduce a two-dimensional model of photorefractive media, with the usual saturable nonlinearity, and a cylindrical photonic lattice periodic in the radial direction. The saturable nonlinearity may be either self-focusing or self-defocusing, and the cylindrical lattice is introduced so that, in the former case, the potential has a maximum at the center. Ring-shaped solitons are constructed numerically. In the case of self-focusing, they are typically unstable against azimuthal perturbations, which is shown by means of numerical computations, and is also explained with the help of an analytical approximation. On the contrary, solitons of the same type may be stable if the saturable nonlinearity is self-defocusing.