Stationary states near the interface with anharmonic properties between linear and nonlinear defocusing media

We consider the nonlinear excitation near the thin layer with nonlinear properties separated linear and media with Kerr-type defocusing nonlinearity. The excitations are described by nonlinear Schrödinger equation with nonlinear potential. The problem is reduced to the solution of linear and nonlinear Schrödinger equations on half spaces with the nonlinear boundary conditions at the interface plane. We propose the generalization of two physical model to describe localized electron phase transition under the framework of Ginsburg-Landau theory and the excitons in a trap under the framework of Bose-Einstein condensation theory. We obtain and analyze the new type nonlinear excitations described by exact solutions of nonlinear Schrödinger equation satisfying the boundary conditions in wide energy range. We derive the energy of localized excitations in explicit form in the long-wave approximation. We calculate the spectral density of stationary states and find its peculiarities. The conditions of existence of localized states are found. The long wave nonlinear excitations can exist for the focusing and defocusing nonlinearity inside the interface.