arXiv subjects

A. I. Korolkov

Publications and source records attributed to A. I. Korolkov.

11 recordsLinked to original sources

arXiv2026

Embedding formulae for diffraction problems on square lattices

We develop embedding formulae for all possible diffraction problems with Dirichlet scatterers on square lattices using the Wiener--Hopf perspective. The embedding formula expresses solutions for arbitrary plane-wave incidence in terms of a finite set of auxiliary problems, eliminating the need to re-solve boundary value problems for each incidence angle. First we derive explicit embedding formulae for canonical geometries including the half-plane, finite strip, and right-angled wedge. We then generalize the method through an operator-based approach, obtaining embedding formula for arbitrary configurations of obstacles on lattices. This general embedding formula is a key difference from the continuous setting where this is currently not possible. To validate the theory, we perform numerical experiments, confirming agreement with the results derived using the embedding formula. The results highlight the efficiency and generality of the Wiener--Hopf approach in discrete diffraction theory, with potential applications in inverse problems and other areas of physics and mathematics.

math-ph↗

arXiv2025

Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. We introduce the universal Riemann domain $\tilde U$, i.e. a sort of ``compactification'' of the universal covering space $\tilde U_2$ over a small tubular neighborhood $N\mathbb{R}^2$ of $\mathbb{R}^2\backslashσ$ in $\mathbb{B}\subset\mathbb{C}^2$, where $\mathbb{B}\subset\mathbb{C}^2$ is a big ball, and $σ$ is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space $\tilde U$ modulo the singularities and the boundary for the standard local degenerations of type $P_1 ,P_2,P_3$ in Pham's [1] notations and for more complicated configurations in $\mathbb{C}^2$. We consider this homology group as a module over the group ring of the $π_1((N\mathbb{R}^2 \cap \mathbb{B})\backslashσ)$ over $\mathbb{Z}$. The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group $H_2(\tilde U_2,\tilde{\partial \mathbb{B}})$) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group $H_2(\tilde U,\tilde U'\cup\tilde{\partial \mathbb{B}})$ (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables.

math-ph↗

arXiv2025

On an analogy between the Wiener--Hopf formulations of discrete and continuous diffraction problems

This article is dedicated to unifying the framework used to derive the Wiener--Hopf equations arising from some discrete and continuous wave diffraction problems.The main tools are the discrete Green's identity and the appropriate notion of discrete normal derivative. The resulting formal analogy between the Wiener--Hopf equations allows one to effortlessly move between the discrete and continuous formulations. The validity of this novel analogy is illustrated through several famous two-dimensional canonical diffraction problems and extended to three-dimensional problems.

math-ph↗

arXiv2024

Recycling solutions of boundary value problems: the Wiener--Hopf perspective on embedding formula

Embedding formula allows to recycle solution of a family boundary value problems by expressing all the solutions in terms of a small number of solutions. Such formulas have been previously derived in the context of diffraction by applying a cleverly chosen operator to the solution and the construction of edge Green's functions which are introduced in an elaborate manner specific for each problem. We demonstrate that embedding formula naturally appears from a matrix Wiener--Hopf equation, and the embedding formula is derived from the canonical solution to this matrix Wiener--Hopf problem. This allows to drive the embedding formula in any context where the problem can be formulated as a Wiener--Hopf equation. We illustrate the effectiveness of this approach by revisiting known problems, such as the problem of diffraction by half-line, a strip and the problem of diffraction by a wedge. Additionally, a new matrix Wiener--Hopf formulation is derived for wedge problems.

math-ph↗

arXiv2021

Diffraction by a Dirichlet right angle on a discrete planar lattice

A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of [1].

math-ph↗

arXiv2021

Saddle point method for transient processes in waveguides

A modification of the saddle point method is proposed for computation of non-stationary wave processes (pulses) in waveguides. The dispersion diagram of the waveguide is continued analytically. A set of possible saddle points on the dispersion diagram is introduced. A method of checking whether the particular saddle points contribute terms to the field decomposition is proposed. A classification of the waveguides based on the topology of the set of possible saddle points is outlined.

physics.comp-ph↗

arXiv2021

Transient processes in a gas / plate structure in the case of light gas loading

Problems of pulse excitation in an acoustic waveguide with a flexible wall and in an acoustic half-space with a flexible wall are studied. In both cases the flexible wall is described by a thin plate equation. The solutions are written as double Fourier integrals. The integral for the waveguide is computed explicitly, and the integral for the half-space is estimated asymptotically. A special attention is paid to the pulse, which is a harmonic wave of a finite duration associated with the coincidence point of the dispersion diagrams of the acoustic medium and the plate. The method of estimating of the double Fourier integral is applied to the problem of excitation of waves in a system composed of an ice plate, water substrate, and the air.

physics.class-ph↗

arXiv2020

Asymptotical study of two-layered discrete waveguide with a weak coupling

A thin two-layered waveguide is considered. The governing equations for this waveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution of this system in the form of a double integral can be obtained by using Fourier transformation. Then, the double integral can be reduced to a single integral with the help of residue integration with respect to the time frequency. However, such an integral can be difficult to estimate since it involves branching and oscillating functions. This integral is studied asymptotically. A zone diagram technique is proposed to represent the set of possible asymptotic formulae. The zone diagram generalizes the concept of far-field and near-field zones.

math-ph↗

arXiv2020

Scalar Klein--Gordon equation and its analytically continued dispersion diagram

The scalar Klein-Gordon equation describes wave motion in a waveguide with a cut-off. For example, the displacement of an elastic cord anchored to a solid base by elastic elements can be described by the scalar Klein-Gordon equation. We analyse this equation using the concept of analytical continuation of dispersion diagram. Particularly, it is shown that the dispersion diagram is topologically equivalent to a tube analytically embedded in two-dimensional complex space. The corresponding Fourier integral is studied on this tube using the Cauchy's theorem. The basic properties of the scalar Klein-Gordon equation are established.

math-ph↗

arXiv2019

Sommerfeld--type integrals for discrete diffraction problems

Three problems for a discrete analogue of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: 1) the problem with a point source on an entire plane; 2) the problem of diffraction by a Dirichlet half-line; 3) the problem of diffraction by a Dirichlet right angle. It is shown that total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As the result, the explicit solutions are obtained in terms of recursive relations (for the Green's function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).

math.NA↗

arXiv2017

Diffraction by an elongated body of revolution. A boundary integral equation based on the parabolic equation

A problem of diffraction by an elongated body of revolution is studied. The incident wave falls along the axis. The wavelength is small comparatively to the dimensions of the body. The parabolic equation of the diffraction theory is used to describe the diffraction process. A boundary integral equation is derived. The integral equation is solved analytically and by iterations for diffraction by a cone.

math.AP↗