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We demonstrate that modulation of the local strength of the cubicself-focusing (SF) nonlinearity in the two-dimensional (2D) geometry, in theform of a circle with contrast $\Delta g$ of the SF coefficient relative to theambient medium with a weaker nonlinearity, stabilizes a family of fundamentalsolitons against the critical collapse. The result is obtained in an analyticalform, using the variational approximation (VA) and Vakhitov-Kolokolov (VK)stability criterion, and corroborated by numerical computations. For the smallcontrast, the stability interval of the soliton's norm scales as $\Delta N\sim\Delta g$ (the replacement of the circle by an annulus leads to a reduction ofthe stability region by perturbations breaking the axial symmetry). To furtherillustrate this mechanism, we demonstrate, in an exact form, the stabilizationof 1D solitons against the critical collapse under the action of a locallyenhanced quintic SF nonlinearity.

Stable two-dimensional solitons supported by radially inhomogeneous self-focusing nonlinearity