Gap solitons in Bragg gratings with dispersive reflectivity

We present a phenomenological generalization of the standard model that describes the propagation of electromagnetic waves in a nonlinear fiber equipped with the Bragg grating (BG). The generalized model includes spatial dispersion of the Bragg reflectivity. The model may apply to nonuniform gratings, including slightly disordered ones. The bandgap remains the same as in the standard model, up to a certain value of the reflectivity-dispersion parameter, and then starts to shrink. Stationary solutions of the model are found numerically. It is found that, above some value of the reflectivity-dispersion parameter, solitons develop well-pronounced sidelobes, which make them drastically different from the classical gap solitons (GSs). The stability region of the GSs in this model is larger than in the standard model. A qualitative explanation to the stabilization of the GSs in the generalized model is proposed. The sidelobes totally alter the character of interactions between the GSs: in-phase solitons repel each other, while out-of-phase ones originally attract each other, but then bounce back.