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Volodymyr Sushch

Publications and source records attributed to Volodymyr Sushch.

At least 19 recordsLinked to original sources
arXiv2025

On a semi-discrete model of Maxwell's equations in three and two dimensions

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell's equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. This approach preserves the intrinsic geometric and topological structures of the continuous theory while providing a consistent spatial discretization. We analyze the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell's equations. As a special case, the model is illustrated on a combinatorial two-dimensional torus, where the semi-discrete Maxwell's equations take the form of a system of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived.

math-ph↗
arXiv2024

2D discrete Yang-Mills equations on the torus

In this paper, we introduce a discretization scheme for the Yang-Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang-Mills equations are presented in the form of both a system of difference equations and a matrix form.

math-ph↗
arXiv2022

2D discrete Hodge-Dirac operator on the torus

We discuss a discretisation of the de Rham-Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

math-ph↗
arXiv2019

Chiral properties of discrete Joyce and Hestenes equations

This paper concerns the question of how chirality is realized for discrete counterparts of the Dirac-Kähler equation in the Hestenes and Joyce forms. It is shown that left and right chiral states for these discrete equations can be described with the aid of some projectors on a space of discrete forms. The proposed discrete model admits a chiral symmetry. We construct discrete analogues of spin operators, describe spin eigenstates for a discrete Joyce equation, and also discuss chirality.

math-ph↗
arXiv2016

Discrete Dirac-Kähler equation and its formulation in algebraic form

A relationship between the discrete Dirac-Kähler equation and discrete analogues of some Dirac type equations in the geometric spacetime algebra is discussed. We show that a solution of the discrete Dirac-Kähler equation can be represented as the sum and difference of solutions of discrete Dirac type equations with the corresponding sign of the mass term.

math-ph↗
arXiv2015

Discrete Dirac-Kähler and Hestenes equations

A discrete analogue of the Dirac equation in the Hestenes form is constructed by introduction the Clifford product on the space of discrete forms. We discuss the relation between the discrete Dirac-Kähler equation and a discrete Hestenes equation.

math-ph↗
arXiv2014

On the chirality of a discrete Dirac-Kähler equation

We discuss a discrete analogue of the Dirac-Kähler equation in which chiral properties of the continual counterpart are captured. We pay special attention to a discrete Hodge star operator. To build one a combinatorial construction of double complex is used. We describe discrete exterior calculus operations on a double comlex and obtain the discrete Dirac-Kähler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac-Kähler equation is shown. Moreover, in the massive case we prove that a discrete Dirac-Kähler operator flips the chirality.

math-ph↗
arXiv2013

A discrete model of the Dirac-Kähler equation

We construct a new discrete analog of the Dirac-Kähler equation in which some key geometric aspects of the continuum counterpart are captured. We describe a discrete Dirac-Kähler equation in the intrinsic notation as a set of difference equations and prove several statements about its decomposition into difference equations of Duffin type. We study an analog of gauge transformations for the massless discrete Dirac-Kähler equations.

math-ph↗
arXiv2011

A double complex construction and discrete Bogomolny equations

We study discrete models which are generated by the self-dual Yang-Mills equations. Using a double complex construction we construct a new discrete analog of the Bogomolny equations. Discrete Bogomolny equations, a system of matrix valued difference equations, are obtained from discrete self-dual equations

math-ph↗
arXiv2009

Self-dual and anti-self-dual solutions of discrete Yang-Mills equations on a double complex

We study a discrete model of the SU(2) Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$. Self-dual and anti-self-dual solutions of discrete Yang-Mills equations are constructed. To obtain these solutions we use both techniques of a double complex and the quaternionic approach. Interesting analogies between instanton, anti-instanton solutions of discrete and continual self-dual, anti-self-dual equations are also discussed.

math-ph↗
arXiv2007

Green function for a two-dimensional discrete Laplace-Beltrami operator

We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one.

math-ph↗
arXiv2006

A gauge-invariant discrete analog of the Yang-Mills equations on a double complex

An intrinsically defined gauge-invariant discrete model of the Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$ is constructed. We develop several algebraic structures on the matrix-valued cochains (discrete forms) that are analogs of objects in differential geometry. We define a combinatorial Hodge star operator based on the use of a double complex construction. Difference self-dual and anti-self-dual equations will be given. In the last section we discus the question of generalizing our constructions to the case of a 4-dimensional combinatorial sphere

math-ph↗
arXiv2004

Discrete model of Yang-Mills equations in Minkowski space

Using methods of differential geometry, a discrete analog of the Yang-Mills equations in Minkowski space is constructed. The gauge transformation law in a discrete formulation is given and gauge invariance of discrete Yang-Mills equations is studied. Difference self-dual and anti-self-dual equations with respect to the Lorentz metric are presented.

math-ph↗