The problem of numerical solution of a nonlinear Schrödinger equation is considered from the point of view of applications to the compensation of signal distortions in a fiber optic communication line. The problem of constructing fast algorithms for the direct and inverse scattering problems for the Zakharov–Shabat system of equations is studied. An overview of the main methods used currently is given. The time complexity of the algorithms is described together with their applicability to realistic signals.

We consider wave propagation across an infinite waveguide of an arbitrary bounded crosssection, whose interior is blocked by two identical thick perforated barriers with holes. When the holes are small, the waves over a broad range of frequencies are almost fully reflected. However, we show the existence of a resonance frequency at which the wave is almost fully transmitted, even for very small holes. Counter-intuitively, this resonance effect occurs for barriers of arbitrary thickness. We also discuss another asymptotic limit, in which the thickness of barriers grows to infinity but the fixed diameter of the holes can be large and even arbitrarily close to the diameter of the waveguide. The resonance scattering, which is known as tunneling effect in quantum mechanics, is demonstrated in a constructive way by rather elementary tools such as separation of variables and matching of the resulting series, in contrast to commonly used abstract methods such as searching for complex-valued poles of the scattering matrix or non-stationary scattering theory. In particular, we derived an explicit equation that determines the resonance frequency. The employed basic tools make the paper accessible to non-experts and educationally appealing. (c) 2021 Elsevier Inc. All rights reserved.

The localization of eigenfunctions of the Laplace operator in a domain divided by a perforated barrier is proven. The localization takes place with sufficiently small holes in the barrier. In this case, the measure of the barrier can be arbitrarily small.

The radiation problem in a regular waveguide with inhomogeneous filling and the scattering problem in a hollow irregular waveguide are considered. The completeness of the system of waveguide root vectors is proven. The radiation conditions are set and the solvability of the excitation problem is established. The dispersion curves of the waveguide with anisotropic filling are analyzed. The singular points of dispersion curves are considered. The excitation problem for a waveguide in a resonance regimen is investigated. The embedding theorems for the vector fields are proven. The solvability of the scattering problem in a waveguide is established.

The frequency dependence of the propagation constants of plane layered dielectric waveguides with the Kerr nonlinearity is considered. An explanation to the possible difference of their behavior from the linear case, related exclusively to a fixed value of an eigenfunction at the boundary of the layer, is given. Explicit formulas for calculating the dispersion curves are obtained. Their behavior for different ways of defining the eigenfunction of the nonlinear problem is analyzed.

Behavior of the dispersion curves of waveguides with anisotropic filling is considered. The existence of backward and complex waves at certain values of the anisotropy coefficients is established. The region of localization of the singular points of the dispersion curves in which the generation of backward waves takes place is found.

We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a "perforated" barrier. We prove that the first eigenfunction is localized in the larger subdomain, i.e., its $L_2$ norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the "holes" in the barrier small enough. This result extends the well known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to noncylindrical domains with radial symmetry. Second, we study a scattering problem in an infinite cylindrical domain with two identical perforated barriers. If the holes are small, there exists a low frequency at which an incident wave is fully transmitted through both barriers. This result is counter-intuitive as a single barrier with the same holes would fully reflect incident waves with low frequences.

The distribution of special points of dispersion curves for an anisotropic filled waveguide is considered. The existence of special curves is substantiated for some relationship between anisotropy coefficients. These points are connected with complex and backward waves. It is proven that the curve of special points is an ellipse.

In this paper, the problem of electromagnetic wave diffraction by extensive conducting bodies with uniform cross sections and continuous curvature boundaries is studied. Corrugated cylinders are considered as diffusers. A resonant decrease in the radiation visibility of such bodies was discovered.

The structure of an operator that determines the partial conditions of radiation in the scalar problem of diffraction theory is considered. Nonlocal boundary conditions are determined by a series setting a certain integro-differential operator. The principal part of this operator is presented in the explicit form of a hyper-singular operator and its components with lower-order singularities. The remaining rapidly converging part of the functional series determines an integral operator with a continuous kernel.

Application of net-point methods to the solution of the layer-to-layer transition problem leads to the necessity to restrict the domain and formulate artificial boundary conditions. In the present work the introduction of an artificial coaxial for reducing the original problem to a waveguide diffraction problem is applied. The test results of the finite-element program that we developed, which demonstrate the high accuracy of the method, are considered. The investigated dependence of the solution on the artificial boundary location shows the rapid convergence of the method.

The localization problem is considered for eigenfunctions of the Laplace operator in a domain that consists of two rectangles linked by a small hole. The localization of the eigenfunction is proven in a subdomain. The velocity is estimated for the convergence of an eigenvalue of the original problem to a subdomain eigenvalue.

We apply the method of mixed finite elements to the junction problem for coaxial and radial waveguides. The problem is reduced to an internal boundary-value problem with nonlocal boundary-value conditions. In the low-frequency range, we compare the results of the finite-element method with the Otto relationship.

The problem of excitation of an anisotropic media-filled waveguide at critical frequencies is considered. An example of a dispersion curve with two rather than one or three singular points is presented. The possibility of excitation of back waves is studied. The character of the increase in the field upon resonance excitation of a waveguide is considered.

We develop a program, which can be used for two-dimensional scalar diffraction problems in a domain with complex structure by finite element method with partial radiation conditions. There are many different ways to solve a diffraction problem. They can be classified in two classes. The first one is the reduction of the boundary value problem to an integral equation. The second way is to solve this boundary value problem directly. The most universal method to solve a boundary value problem is the finite element method.