LOCALIZED STATES ON THE BOUNDARY BETWEEN LINEAR AND NONLINEAR MEDIA

Abstract

The paper presents a model describing the peculiarities of excitation localization on the boundary between linear and nonlinear media. 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 linear and nonlinear Schrödinger equations on half-spaces. Explicit solutions of nonlinear Schrödinger equations satisfying the boundary conditions were found for positive and negative nonlinearity parameters. It was demonstrated that nonlinear localized excitations can be of several types. The structure and the shape of the localized states are determined by the type of anharmonicity of the interaction in the medium and the intensity of the interaction between the excitations and the defect. There are two types of states: localized and quasilocal. Localized states are described by wave functions that damp away from the interface of media under various laws. The profile of such a bound state is asymmetric in relation to the media interface. Quasi-local states are described by wave functions monotonically damped in a half-space with a nonlinear medium and a standing wave in a half-space with a linear medium. The wave is localized during the transition through the medium interface. The dispersion relations determining the values of the energy of localized and quasilocal states were obtained. The dependence of the wave numbers on the parameters of the system for localized states in various limiting cases was demonstrated. The additions to the spectral density of quasilocal states were derived.