Interaction of a soliton with a localized gain in a fiber Bragg grating

A model of a lossy nonlinear fiber grating with a hot spot, which combines a local gain and an attractive perturbation of the refractive index, is introduced. A family of exact solutions for pinned solitons is found in the absence of loss and gain. In the presence of the loss and localized gain, an instability threshold of the zero solution is found. If the loss and gain are small, it is predicted what soliton is selected by the energy-balance condition. Direct simulations demonstrate that only one pinned soliton is stable in the conservative model, and it is a semiattractor: solitons with a larger energy relax to it via emission of radiation, while those with a smaller energy decay. The same is found for solitons trapped by a pair of repulsive inhomogeneities. In the model with the loss and gain, stable pinned pulses demonstrate persistent internal vibrations and emission of radiation. If these solitons are nearly stationary, the prediction based on the energy balance underestimates the necessary gain by 10-15% (due to radiation loss). If the loss and gain are larger, the intrinsic vibrations of the pinned soliton become chaotic. The local gain alone, without the attractive perturbation of the local refractive index, cannot maintain a stable pinned soliton. For collisions of moving solitons with the hot spot, passage and capture regimes are identified, the capture actually implying splitting of the soliton.