Using the concept of natural geometry in the nonlinear electrodynamics of the vacuum

We discuss the concept of natural geometry for a physical field. We show that using this concept to solve problems of nonlinear electrodynamics allows simply studying basic effects of nonlinear electrodynamics arising when weak electromagnetic waves propagate in external electromagnetic fields.     

Nonlinear electrodynamic lensing of electromagnetic waves in a magnetic dipole field

We consider propagation of electromagnetic waves in magnetic dipole and gravitational fields proceeding in accordance with the nonlinear vacuum electrodynamics laws. We derive formulas describing the effect of nonlinear electrodynamic lensing of electromagnetic waves in the magnetic dipole field. We show that rotation of the magnetic dipole moment about an axis noncoincident with this moment leads to a nonlinear electrodynamic modulation of the electromagnetic radiation intensity by frequencies that are multiples of the dipole rotation frequency. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 85–94, January, 2007.     

Interaction Effect of Plane Electromagnetic Waves in the Born–Infeld Nonlinear Electrodynamics

We calculate the interaction of plane electromagnetic waves in accordance with the Born–Infeld nonlinear electrodynamics in the vacuum segment of a ring laser. We show that at the present level of development in laser physics, it is possible not only to measure the direct impact of the quantum vacuum polarization effect on the generation frequencies of weak electromagnetic waves propagating in opposite directions in a ring laser but also to estimate the contribution of the Born–Infeld nonlinear electrodynamics to this effect.     

Investigation of the Effective Space–Time of the Vacuum Nonlinear Electrodynamics in a Magnetic Dipole Field

The metric tensor of the effective pseudo-Riemannian space–time for an electromagnetic wave propagating in the magnetic dipole field and the gravitational field of a neutron star is obtained within a parameterized post-Maxwellian vacuum electrodynamics. The angles of the nonlinear electrodynamic and gravitational ray bending for electromagnetic waves propagating in the magnetic equatorial plane of the star are calculated based on an analysis of isotropic geodesics of this space. We show that for all nonlinear theories whose post-Maxwellian parameters do not coincide, the velocity of the electromagnetic signal propagation in external fields and the rays along which these signals propagate depend on the polarization of the electromagnetic waves. The difference of the source-to-detector propagation time of these signals for two principal polarization states is calculated.     

Development of the concept of natural geometry for physical interactions

We analyze the development of the concept of natural geometry for the gravitational field in Logunov’s works. We discuss the application of this concept to vacuum nonlinear electrodynamics and show that defining the natural geometry for a nonlinear theory and finding its metric tensor permit obtaining sufficiently complete information about the propagation laws for electromagnetic field pulses in background electromagnetic fields.     

Nonlinear Electrodynamic Delay of Electromagnetic Signals in a Coulomb Field

We study the motion of electromagnetic signals in the Coulomb electric field of a compact gravitating center under the laws of the parameterized nonlinear electrodynamics of the vacuum. We evaluate the deflection angles of the rays along which the electromagnetic waves of two normal modes propagate in this field. We determine the nonlinear electrodynamic delay of the electromagnetic signal of one normal mode relative to the electromagnetic signal of another normal mode during their propagation from the same source to an observer.     

Mathematical modeling of electromagnetic wave propagation in nonlinear electrodynamics

The eikonal method for an electromagnetic wave propagating according to the laws of non-linear electrodynamics in vacuum in external electromagnetic and gravitational fields is developed. A mathematical model of the propagation of electromagnetic signals in the parameterized post-Maxwellian electrodynamics in vacuum is constructed. As an example of using the proposed method, the angles of the nonlinear electrodynamical and gravitational curvature of the normal wave rays propagating in the field of a charged collapsar are calculated.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

An initial boundary-value problem for model electromagnetoelasticity system

In this paper we consider an initial boundary-value problem related to the electrodynamics of vibrating elastic media. The aim is to prove an existence and uniqueness result for a model describing the nonlinear interactions of the electromagnetic and elastic waves. We assume that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations, one of them is the hyperbolic equation (an analog of the Lamé system) and another one is the parabolic equation (an analog of the diffusion Maxwell system). One stability result is proved too.     

Basic systems of orthogonal functions for space-time multivectors

Space-time multivectors in Clifford algebra (space-time algebra) and their application to nonlinear electrodynamics are considered. Functional product and infinitesimal operators for translation and rotation groups are introduced, where unit pseudoscalar or hyperimaginary unit is used instead of imaginary unit. Basic systems of orthogonal functions (plane waves, cylindrical, and spherical) for space-time multivectors are built by using the introduced infinitesimal operators. Appropriate orthogonal decompositions for electromagnetic field are presented. These decompositions are applied to nonlinear electrodynamics. Appropriate first order equation systems for cylindrical and spherical radial functions are obtained. Plane waves, cylindrical, and spherical solutions to the linear electrodynamics are represented by using the introduced orthogonal functions. A decomposition of a plane wave in terms of the introduced spherical harmonics is obtained.     

Maxwell equations in complex form of Majorana–Oppenheimer,solutions with cylindric symmetry in Riemann <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript> and Lobachevsky <Emphasis Type="Italic">H</Emphasis><Subscript>3</Subscript> spaces

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode–Weyl–Fock–Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for the Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.     

Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green’s tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.     

Riemann-Cartan-Weyl geometries, quantum diffusions and the equivalence of the free maxwell and Dirac-Hestenes equations

We introduce the Riemann-Cartan-Weyl (RCW) space-time geometries of quantum mechanics with the most general trace-torsion non-exact Weyl 1-form, and characterize it in the Clifford bundle. Two electromagnetic potentials appear in the Weyl form, one having a zero field and the other one being the codifferential of a 2-form. We give the derivation of the non-linear equation for the wave function producing the exact Weyl one-form, which also defines the amplitude of a Dirac-Hestenes spinor operator field (DHSOF). We prove an equivalence between the free Maxwell equation for an extremal electromagnetic field and the Dirac-Hestenes equation for a DHSOF on a Riemann-Cartan-Weyl manifold, associating the electromagnetic potentials of the Weyl one-form with the internal electromagnetic potentials derived from the rotational dependance of a DHSOF. We show that this association produces a breaking of detailed balance in the spin plane. We discuss the relations with stochastic electrodynamics and the Navier-Stokes equation.     

On a group approach to studying the Einstein and maxwell equations

Based on the classical problem for decomposition of the tensor product of representations into irreducible components, which is considered in the elementary representation theory for orthogonal groups, a partial classification of the Einstein equations is carried out. A new class of Maxwell equations for relativistic electrodynamics is singled out and studied. Pointwise-irreducible decompositions for the energy-momentum and electromagnetic field tensors are obtained and a physical interpretation of the decomposition components is given. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 32–43, April, 1997.     

Dynamical systems and simulation of turbulence

We propose an approach to the analysis of turbulent oscillations described by nonlinear boundary-value problems for partial differential equations. This approach is based on passing to a dynamical system of shifts along solutions and uses the notion of ideal turbulence (a mathematical phenomenon in which an attractor of an infinite-dimensional dynamical system is contained not in the phase space of the system but in a wider functional space and there are fractal or random functions among the attractor “points”). A scenario for ideal turbulence in systems with regular dynamics on an attractor is described; in this case, the space-time chaotization of a system (in particular, intermixing, self-stochasticity, and the cascade process of formation of structures) is due to the very complicated internal organization of attractor “points” (elements of a certain wider functional space). Such a scenario is realized in some idealized models of distributed systems of electrodynamics, acoustics, and radiophysics. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 217–230, February, 2007.     

Some aspects of applying polynorms in field theory

We consider some possibilities of physical applications of pseudonorms of an order higher than two (polynorms) in hypercomplex algebras, primarily in the biquaternion algebra. We can then view several known questions from a new standpoint. In particular, we show that considering the 4-norm in field theory ensures a natural transition from the Maxwell electrodynamics to the nonlinear Born-Infeld electrodynamics. Moreover, the algebraic approach shows that it is natural to add the Skyrme nonlinear term to the meson Lagrangian of nuclear forces. We also find that the only fourth-order additional term can be naturally added to the Skyrme Lagrangian, which might improve the model properties.     

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.     

Threshold Phenomena in Intense Electromagnetic Fields

The influence of intense electromagnetic fields on the formation and decay of quasistationary states of different quantum systems is investigated based on exact solutions of quantum equations for charged particle motion. The method allows examining systems where a spontaneous decay may occur as well as phenomena that occur only under the action of the field. Different values of the total magnetic moment of the system are taken into account in this consideration. A consistent use of the analytic continuation method allows obtaining nonlinear equations that determine complex energies in an external field. The asymptotic expansions for real and imaginary energy values under the action of weak and strong electromagnetic fields are investigated. The developed approach allows establishing the characteristic values for the length parameters that determine the formation of the processes in superstrong fields. We note that a significant decrease of distances in strong fields may lead to effects with a new characteristic length scale, characterizing a modified quantum electrodynamics (QED) formalism, namely, the QED with the fundamental mass formalism.     

Solutions to nonlinear Schr?dinger equations with singular electromagnetic potential and critical exponent

We investigate existence and qualitative behavior of solutions to nonlinear Schr?dinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of degree –1, including the Aharonov–Bohm class. In particular, by variational arguments we prove a result of multiplicity of solutions distinguished by symmetry properties.