Spatial solitons supported by localized gain in nonlinear optical waveguides

We introduce a modification of the complex Ginzburg-Landau (CGL) equation with background linear loss and locally applied gain. The equation appertains to laser cavities based on planar waveguides, and also to the description of thermal convection in binary fluids. With the gain localization accounted for by the delta-function, a solution for pinned solitons is found in an analytical form, with one relation imposed on parameters of the model. The exponentially localized solution becomes weakly localized in the limit case of vanishing background loss. Numerical solutions, with the delta-function replaced by a finite-width approximation, demonstrate stability of the pinned solitons and their existence in the general case, when the analytical solution is not available. If the gain-localization region and the size of the soliton are comparable, the static soliton is replaced by a stable breather.