Diffraction of the Fundamental Waveguide Wave on an Inductive Cylinder in a Rectangular Waveguide

Computational Mathematics and Modeling - We apply the integral equation method to the diffraction of a waveguide wave on an impedance inductive cylinder in a rectangular waveguide. The...     

A Canonical Diffraction Problem With Two Media

A Wiener–Hopf equation in L² being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ=ℝ₊×{0} which faces two different media Ω_-: x₂<0, Ω₊: x₂>0, as a special configuration in [3], is solved by canonical Weiner–Hopf factorization of its L²-regular scalar symbol γ_o=γ_o- γ_o+. The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi-infinite branch cuts of t_j(ξ):=(ξ²−k²_j)^1/2, k_j∈ℂ⁺⁺ for j=1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.     

On a Nonlinear Spectral Problem for a Dielectric Waveguide with Kerr Nonlinearity

Computational Mathematics and Mathematical Physics - The frequency dependence of the propagation constants of plane layered dielectric waveguides with the Kerr nonlinearity is considered. An...     

Integral Dispersion Equation Method in the Problem on Nonlinear Waves in a Circular Waveguide

Differential Equations - We study TE-polarized electromagnetic waves propagating in an inhomogeneous dielectric waveguide of circular cross-section filled with a nonlinear medium where the...     

Diffraction Approach in the Scattering Problem for Three Charged Quantum Particles

Mathematical Notes -     

On an Integral Equation in the Problem of Diffraction of a Plane Wave on a Transparent Circular Cone

The problem of diffraction on a transparent convex cone is studied. A uniqueness theorem is proved for the case where the cone is illuminated by a compact source. For a circular cone, the solution is obtained in the form of Kontorovich-Lebedev integrals and Fourier series expansions. A singular integral equation is deduced for the Fourier coefficients, and its regularization is carried out. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 101–123.     

Solution of a Boundary-Value Problem associated with Diffraction of Water Waves by a Nearly Vertical Barrier

An exact solution is derived for a boundary-value problem forLaplace's equation which is a generalization of the one occurringin the course of solution of the problem of diffraction of surfacewater waves by a nearly vertical submerged barrier. The methodof solution involves the use of complex function theory, theSchwarz reflection principle, and reduction to a system of twouncoupled Riemann-Hilbert problems. Known results, representingthe reflection and transmission coefficients of the water waveproblem involving a nearly vertical barrier, are derived interms of the shape function.     

Operator Function Method in the Problem of Normal Waves in an Inhomogeneous Waveguide

The problem of normal waves in a closed (screened) regular waveguiding structure of arbitrary cross-section is considered by reducing it to a boundary value problem for the longitudinal electromagnetic field components in Sobolev spaces. The variational statements of the problem is used to determine the solution. The problem is reduced to studying an operator function. The properties of the operators contained in the operator function necessary to analyze its spectral properties are studied. Theorems on the spectrum discreteness and the distribution of characteristic numbers of the operator function on the complex plane are proved. The problem of completeness of the system of root vectors of the operator function is considered. The theorem on the double completeness of the system of root vectors of the operator function with finite deficiency is proved.     

The Diffraction Problem and Verigin Problem of Quasilinear Parabolic Equation in Divergence Form for the One-dimensional Case

In this paper, we consider the flow of two immiscible fluids in a onedimensional porous medium (the Verigin problem) and obtain a quasilinear parabolic equation in divergence form with the discontinuous coefficients. We prove first the existence and uniqueness of locally classical solution of the diffraction problem and then prove the existence of local solution of the Verigin problem.     

Scattering in a Forked-Shaped Waveguide

We consider wave scattering in a forked-shaped waveguide which consists of two finite and one half-infinite intervals having one common vertex. We describe the spectrum of the direct scattering problem and introduce an analogue of the Jost function. In case of the potential which is identically equal to zero on the half-infinite interval, the problem is reduced to a problem of the Regge type. For this case, using Hermite-Biehler classes, we give sharp results on the asymptotic behavior of resonances, that is, the corresponding eigenvalues of the Regge-type problem. For the inverse problem, we obtain sufficient conditions for a function to be the S-function of the scattering problem on the forked-shaped graph with zero potential on the half-infinite edge, and present an algorithm that allows to recover potentials on the finite edges from the corresponding Jost function. It is shown that the solution of the inverse problem is not unique. Some related general results in the spectral theory of operator pencils are also given. This work was supported by the grant UM1-2567-OD-03 from the Civil Research and Development Foundation (CRDF). YL was partially supported by the NSF grants 0338743, 0354339 and 0754705, by the Research Board and Research Council of the University of Missouri, and by the EU Marie Curie “Transfer of Knowledge” program.     

On the Solvability of the Problem of Electromagnetic Wave Diffraction by a Layer Filled with a Nonlinear Medium

Computational Mathematics and Mathematical Physics - The problem of diffraction of a polarized electromagnetic wave by a layer filled with a nonlinear medium is considered. The layer is located...     

Existence of a Solution of the Inverse Problem Estimating the Structure of Materials from Diffraction Data

The first part of the article proves the existence of a solution for the problem of estimating the full set of interatomic distances, which easily leads to determination of single crystal structure. The second part proves the existence of a solution for the problem of estimating the density of a spherically symmetrical particle from small-angle scattering data.     

Diffraction in a nonlinear defocusing medium

Diffraction by a semitransparent screen is considered in the framework of the nonlinear Schrödinger equation. The case of a defocusing medium is investigated. The method used represents a synthesis of the method of Riemann's matrix problem and the technique of Whitham's deformations of spectral curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 23–31, 1989.     

A Mushy Region in a Stefan Problem

In the melting of a pure metal by volume heating, when superheatingdoes not occur, there will be a region in which the metal isneither pure liquid nor pure solid, called a mushy region. Agrain or dendrite model is proposed to describe the microstructureof the metal in this situation, and its stability is discussed.The explicit solution for the one-dimensional version of themodel is obtained and has a different form in three regions;near the pure solid boundary, in the mush, and near the pureliquid boundary. An appropriate average of this solution forthe microstructure is shown to reduce to the weak solution forthe microstructure is shown to reduce to the weak solution forthe macroscopic problem proposed by Atthey (1974). Propertiesof the solution for more general grain geometries are also discussed.     

Inverse Problems for Obstacles in a Waveguide

In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ?X. In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry.