PERIODIC STATES NEAR THE PLANE DEFECT WITH NON-LINEAR RESPONSE SEPARATING NON-LINEAR SELF-FOCUSING AND LINEAR CRYSTALS

Abstract

The paper presents a model describing the peculiarities of localised excitation on the boundary between linear and nonlinear self-focusing media. The medium boundary is a plane defect with internal nonlinear properties. The boundary of nonlinear media, characterised by various parameters of anharmonicity of the interatomic interaction, creates a disturbance of the medium characteristics, which is located at distances much smaller than the width of the localization of propagating waves. The model is based on the nonlinear Schrödinger equation with a nonlinear self-consistent potential. The problem is reduced to the solution of the linear and nonlinear Schrödinger equations on half-spaces coupled by the boundary conditions. The nonlinearity in the Schrödinger equation is assumed to be of the Kerr type with a positive parameter. Explicit solutions of nonlinear Schrödinger equations satisfying the boundary conditions were found for positive and negative nonlinearity parameters. It is shown that the existence of nonlinear spatially inhomogeneous states of several types determined by various periodic solutions of the nonlinear Schrödinger equation is possible in the system under consideration. The dispersion relations determining the energy of such stationary states were obtained and analysed. The energy dependences on the system parameters for stationary states in various limiting cases were obtained in an explicit form. It was established that resonance states exist in the spectrum, determined exclusively by the nonlinear properties of the defect. The additions to the spectral density of states were obtained, and its characteristic features were determined.