Stable two-dimensional solitons supported by radially inhomogeneous   self-focusing nonlinearity
  We demonstrate that modulation of the local strength of the cubic self-focusing (SF) nonlinearity in the two-dimensional (2D) geometry, in the form of a circle with contrast $\Delta g$ of the SF coefficient relative to the ambient medium with a weaker nonlinearity, stabilizes a family of fundamental solitons against the critical collapse. The result is obtained in an analytical form, using the variational approximation (VA) and Vakhitov-Kolokolov (VK) stability criterion, and corroborated by numerical computations. For the small contrast, 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 of the stability region by perturbations breaking the axial symmetry). To further illustrate this mechanism, we demonstrate, in an exact form, the stabilization of 1D solitons against the critical collapse under the action of a locally enhanced quintic SF nonlinearity. 
Hidetsugu Sakaguchi and Boris A. Malomed
nlin.PS