INTERACTION OF THE LOCALIZED STATES NEAR NONLINEAR MEDIA BORDER
In this paper the coupled states that are localized near the nonlinear boundary media and propagating along it are considered. The boundary of nonlinear media with different parameters of anharmonicity of interatomic interaction creates a disturbance of medium characteristics, which is focused at distances much smaller than the width of the localization of the propagating waves. It is expected that the particle has a complex linear law of dispersion with several branches different parameters in a model proposed in this paper. The problem is reduced to the solution of nonlinear Schrödinger equations with boundary conditions of a special kind. Explicit solutions of nonlinear Schrödinger equations with positive nonlinear parameters satisfying the boundary conditions were found.
The relations which at a fixed energy allow determining the directions of the wave vector of the wave localized near the boundary were derived
It is shown that taking into account the interaction of the branches of the dispersion law at the boundary leads to the modification of the domain of existence of nonlinear localized waves. The possible existence of nonlinear localized states which are asymmetric in relation to the media boundary was founded. These states are described by wave functions dampening moving to a distance from the boundary. They have a soliton-like profile in the direction perpendicular to the boundary. The structure and shape of such states are determined by the sign of interaction anharmonism in the medium and by the intensity of the excitation interaction and other characteristics of the medium. The equations determining the energy of the wave localized along the media boundary for a fixed direction of its wave vector are derived. Dependences of the wave numbers from the parameters of the system for localized states in various private cases are explicitly expressed. A two-dimensional case determined by a fixed value of the wave vector direction along the boundary of nonlinear medium is also considered. It is shown that if to select a control parameter the value of the interaction energy intensity at the boundary then at its changing in the system of parameters one can get different types of localized states of an asymmetric type.
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