A theoretical study of the features of the localization of fields (light beam) near a plane defect in media with a stepwise defocusing nonlinearity of the Kerr type is carried out. A one-dimensional model is used based on a generalization of the nonlinear Schrödinger equation with a point potential that simulates the interaction of excitations with a plane defect, and a nonlinear term in which the coefficients of the linear and nonlinear responses abruptly change in depending on the field amplitude in the medium. It is found two types of symmetric localized stationary states in different energy ranges. The influence of the intensity of the interaction of excitations with a defect on the profile of localization of states and the conditions for their existence is studied.
The main design of this paper is to adopt potential functions for solving plane defect problems originating from two-dimensional decagonal quasicrystals. First, we analyze the strict potential function theory for the plane problems of two-dimensional quasicrystals. To clarify effectiveness of the method, we give some examples and the results which can be precisely determined, including the elasticity and fracture theories of two-dimensional quasicrystals. These results maybe play a positive role in studying the fracture of two-dimensional quasicrystals in the future.
We analyze guided waves in the linear media separated nonlinear interface. The mathematical formulation of the model is a one-dimensional boundary value problem for the nonlinear Schrödinger equation. The Kerr type nonlinearity in the equation is taken into account only inside the waveguide. We show that the existence of nonlinear stationary waves of three types is possible in defined frequency ranges. We derive the frequency of obtained stationary states in explicit form and find the conditions of its existence. We show that it is possible to obtain the total wave transition through a plane defect. We determine the condition for realizing of such a resonance. We obtain the reflection and transition coefficients in the vicinity of the resonance. We establish that complete wave propagation with nonzero defect parameters can occur only when the nonlinear properties of the defect are taken into account.
In this paper, the new type of coupled states 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 characteristic. It is expected that the particle has a complex linear law of dispersion with several branches of different parameters in a model proposed in this paper. The problem is reduced to the solution of the nonlinear Schrödinger equation with boundary conditions for a special kind. Explicit solutions of nonlinear Schrödinger equations satisfying the boundary conditions were found. It is shown that the existence of nonlinear localized excitations of several types is possible. They have a soliton-like profile in the direction perpendicular to the boundary. The structure and shape of the localized states is determined by the anharmonicity parameters and the intensity of interaction of the excitations with the plane defect. 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.
Особенности формирования локализованных состояний вблизи плоского дефекта в средах со скачкообразно меняющимся дефокусирующим нелинейным откликом
