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We introduce a dynamical model of a Bose-Einstein condensate based on the 2DGross-Pitaevskii equation, in which the nonlinear coefficient is a function ofradius. The model describes a situation with spatial modulation of the negativeatomic scattering length, via the Feshbach resonance controlled by a properlyshaped magnetic of optical field. We focus on the configuration with thenonlinear coefficient different from zero in a circle or annulus, including thecase of a narrow ring. Two-dimensional axisymmetric solitons are found in anumerical form, and also by means of a variational approximation; for aninfinitely narrow ring, the soliton is found in an exact form (in the lattercase, exact solitons are also found in a two-component model). A stabilityregion for the solitons is identified by means of numerical and analyticalmethods. In particular, if the nonlinearity is supported on the annulus, theupper stability border is determined by azimuthal perturbations; the stabilityregion disappears if the ratio of the inner and outer radii of the annulusexceeds a critical value 0.47. The model gives rise to bistability, as thestationary solitons coexist with stable axisymmetric breathers, whose stabilityregion extends to higher values of the norm than that of the static solitons.The collapse threshold strongly increases with the radius of the inner hole ofthe annulus. Vortex solitons are found too, but they are unstable.Comment: 11 pages, 9 figure

Two-dimensional solitons in the Gross-Pitaevskii equation with spatially modulated nonlinearity