Localization of excitations near a thin defect layer with nonlinear properties separating linear and nonlinear crystals

It is shown that localized and quasi-local states exist near a thin defect layer with nonlinear properties, separating a linear medium from a Kerr-type nonlinear medium. Localized states are characterized by a monotonically decreasing field amplitude on both sides of the interface between the media. Quasi-local states are described by the field in the form of a standing wave in the linear medium and a monotonically decreasing field in the nonlinear medium. Contacts with nonlinear self-focusing and defocusing media are analyzed. The mathematical formulation of the proposed model is a system of linear and nonlinear Schrödinger equations with a potential simulating the thin defect layer, which is nonlinear relative to the field. Dispersion relations determining the energy of local and quasi-local states are obtained. The expressions for energy in explicit analytic form are indicated in the limiting cases and the conditions of their existence.