Том 243, № 5 (2019)
- Год: 2019
- Статей: 16
- URL: https://journals.rcsi.science/1072-3374/issue/view/15039
Article
Alexander Pavlovich Kachalov March 27, 1949 – October 1, 2017
A Point Source of Electromagnetic Waves in an Inhomogeneous Medium: a High Frequency Ansatz and the Dual Nonstationary Singular Solution
Аннотация
The Maxwell equations for smoothly inhomogeneous media are addressed. The first-order term of the Hadamard-type expansion for the wavefield of a pulsed point source is presented. The Fourier transform in t of this expression allows for a high-frequency asymptotic formula for the wavefield of a point source of electromagnetic oscillations.
The Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum. The Scattering Problem of Three One-Dimensional Quantum Particles
Аннотация
In the paper the asymptotic structure of eigenfunctions of the absolutely continuous spectrum of the scattering problem is described. The case of three one-dimensional quantum particles interacting by repulsive pair potentials with a compact support is considered.
Simplest Test for the Two-Dimensional Dynamical Inverse Problem (BC-Method)
Аннотация
The dynamical system
is under consideration, where \( {\mathbb{R}}_{+}^2:= \left\{\left(x,y\right)\in {\mathbb{R}}^2\left|y\right.>0\right\} \); ρ = ρ(x, y) is a smooth positive function; f = f(x, t) is a boundary control; u = uf(x, y, t) is a solution. With the system one associates a response operator \( R:f\mapsto {u}^f\left|{}_{y=0}\right. \). The inverse problem is to recover the function ρ via the response operator. A short presentation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided.
If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way of making use of them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.
On Waves Generated by Sources Localized at Infinity
Аннотация
The space-time ℝ4 is compactified by adding the manifold of infinitely distant points. The problem of constructing a solution of the wave equation with the right-hand side (the source of waves), which is a generalized function with support on the manifold of infinitely distant points, is posed and solved. Strict necessary and sufficient conditions that the source must satisfy are stated.
On the Bateman–Hörmander Solution of the Wave Equation Having a Singularity at a Running Point
Аннотация
Hörmander has presented a remarkable example of a solution of the homogeneous wave equation, which has a singularity at a running point. An analytic investigation of this solution is performed for the case of three spatial variables. The support of this solution is described, its behavior near the singular point is studied, and its local integrability is established. It is observed that the Hörmander solution is a specialization of a solution found by Bateman five decades in advance.
Electromagnetic Waves Scattering on an Array Composed of Thin Dielectric Objects
Аннотация
This research concerns the scattering of electromagnetic waves by thin dielectric impediments in 2D geometry. Dielectric and geometric properties of the impediments are modeled by varying the inhomogeneous component of the refractive index. It is assumed that the impediments have arbitrary finite lengths, and their widths are much lesser in comparison with the incident wavelength. In contrast to the previous approaches the proposed one enables us to solve the scattering problem simultaneously on several objects in the case where the impediments structure is not a regular one. A system of integral equations to provide a solution to the problem is derived. A unique solvability of the obtained system is discussed.
On the Cauchy Problem for the Wave Equation with Data on the Boundary
Аннотация
The Cauchy problem for the wave equation in Ω × ℝ with data given on some part of the boundary ∂Ω × ℝ is considered. A reconstruction algorithm for this problem based on analytic expressions is given. This result is applicable to the problem of determining a nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.
“Separation of Variables” in the Model Problems of the Diffraction Theory. A Formal Scheme
Аннотация
The parabolic equation describes the propagation of localized waves along the boundary with singularities. Some reformulation of the “separation of variables” scheme is presented, which enables us to obtain a rich set of solutions of the corresponding boundary value problems.
The One-Dimensional Inverse Problem in Photoacoustics. Numerical Testing
Аннотация
The problem of reconstruction of the Cauchy data for the wave equation in ℝ1 from the measurements of its solution on the boundary of a finite interval is considered. This is a one-dimensional model for the multidimensional problem of photoacoustics, which was studied previously. The method was adapted and simplified for the one-dimensional situation. The results of numerical testing to see the rate of convergence and the stability of the procedure are given. Some hints are also given on how the procedure of reconstruction can be simplified in the 2d and 3d cases.
Green’s Function for the Helmholtz Equation in a Polygonal Domain of Special Form with Ideal Boundary Conditions
Аннотация
A formal approach for the construction of the Green’s function in a polygonal domain with the Dirichlet boundary conditions is proposed. The complex form of the Kontorovich–Lebedev transform and the reduction to a system of integral equations is employed. The far-field asymptotics of the wave field is discussed.
Asymptotics of Eigenvalues in Spectral Gaps of Periodic Waveguides with Small Singular Perturbations
Аннотация
The asymptotics of eigenvalues appearing near the lower edge of a spectral gap of the Dirichlet problem is studied for the Laplace operator in a d-dimensional periodic waveguide with a singular perturbation of the boundary by creating a hole with a small diameter ε. Several versions of the structure of the gap edge are considered. As usual, the asymptotic formulas are different in the cases d ≥ 3 and d = 2, where the eigenvalues occur at distances O(ε2(d−2)) or O(ε2d) and O(|ln ε|−2) or O(ε4), respectively, from the gap edge. Other types of singular perturbation of the waveguide surface and other types of boundary conditions are discussed, which provide the appearance of eigenvalues near both edges of one or several gaps.
On the Morse Index for Geodesic Lines on Smooth Surfaces Embedded in ℝ3
Аннотация
The paper is devoted to the calculation of the Morse index on geodesic lines upon smooth surfaces embedded into the 3D Euclidean space. The interest in this theme is created by the fact that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise. Two types of geodesic flows are considered upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. The position of the points where geodesic lines meet caustics, i.e., focal points, is found and it is proved that all focal points are simple (not multiple) irrespective of the geometric structure of the caustics arisen. The mathematical techniques in use are based on the complexification of the geometrical spreading problem for a geodesics/rays tube.
The Wave Model of the Sturm–Liouville Operator on an Interval
Аннотация
In the paper the wave functional model of a symmetric restriction of the regular Sturm-Liouville operator on an interval is constructed. The model is based upon the notion of the wave spectrum and is constructed according to an abstract scheme, which was proposed earlier. The result of the construction is a differential operator of the second order on an interval, which differs from the original operator only by a simple transformation.
On Adiabatic Normal Modes in a Wedge-Shaped Sea
Аннотация
A two-dimensional problem that is a model for sound propagation in a narrow water wedge near the shore of a sea is studied. A solution to the Helmholtz equation, which is asymptotically a normal wave propagating along “water” wedge to the “shore,” is constructed explicitly. The solution satisfies the Helmholtz equation in the quadrant one side of which is “the surface of the water” and the second is perpendicular to it, starts at the top of the wedge and goes into the “bottom.” Boundary conditions on wedge boundaries and at infinity in the “bottom” are satisfied.