Solitons supported by a self-defocusing trap in a fractional-diffraction waveguide

We introduce a model which gives rise to self -trapping of fundamental and higher -order localized states in a one-dimensional nonlinear Schr & ouml;dinger equation with fractional diffraction and the strength of the self -defocusing nonlinearity growing steeply enough from the center to periphery. The model can be implemented in a planar optical waveguide. Stability regions are identified for the fundamental and dipole (single -node) states in the plane of the L & eacute;vy index and the total power (norm), while states of higher orders are unstable. Evolution of unstable states is investigated too, leading to spontaneous conversion towards stable modes with fewer node.