Field sensitivity to variations of a scatterer

For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational (Fréchet) derivative of the scattered field measured in the H¹ (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<∞, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L^p.     

Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Ionization in a two-mode quantized electromagnetic field

We find an analytic solution of the Schrödinger equation for an electron in a two-mode quantized electromagnetic field. The obtained solution allows calculating the spectra of photoelectrons and atom ionization rates in strong electromagnetic fields.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

Temperature fields and stresses in a system of plane-parallel layers under heating by electromagnetic radiation

We determine and study the temperature fields and stresses arising in heating of a system of two plane-parallel layers of different transparency by electromagnetic radiation. We discuss the possibility of using the results obtained to develop rational programs for thermal processing of electrovacuum devices with a glass shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 21–26.     

Numerical solution of the nonlinear problem of motion of a long rectangular plate in a variable magnetic field

The coupled problem of motion of a long rectangular plate in a variable magnetic field is analyzed in the framework of the geometrically nonlinear theory of thin shells. The proposed numerical procedure is applied to estimate the effect of external electromagnetic and mechanical fields on the stress- strain state of the plate.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 37–42, 1990.     

On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.     

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.     

Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.     

Wave function and the probability current distribution for a bound electron moving in a uniform magnetic field

We study the effects of electromagnetic fields on nonrelativistic charged spinning particles bound by a short-range potential. We analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional δ-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependences of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.     

Local free boundary problem for viscous compressible magnetohydrodynamics

We consider the motion of viscous compressible magnetohydrodynamics fluid in a domain bounded by a free surface. In the external domain, there is electromagnetic field generated by some currents that keeps the magnetohydrodynamics flow in the bounded domain. Then on the free surface, transmission conditions for electromagnetic fields are imposed. In this paper, we prove the existence of local regular solutions by the method of successive approximations. The L₂ approach is used. This helps us to treat the transmission conditions.     

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.     

Rational regimes for induction heating of cylindrical articles for graphitation

We develop a method of computing regimes for induction heating of cylindrical carbon devices during graphitation. We study the electromagnetic, temperature, and mechanical fields as functions of the parameters of the heating regime.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 27–31.     

Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term ${{\bf A} \wedge {\it d}{\bf A}}$ that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.     

Stability of multiply connected ribbed shells and plates in a magnetic field

We explain a method of studying electromagnetic loads, the stress-strain state, and the stability of multiply connected ribbed plates and shells made up of conducting materials and subject to the action of a variable magnetic field. We give the solutions of test problems. We study the influence of ribbing on the magnitude of the critical loads for various multiply connected plates.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 83–89.     

Determination of Space-Time Structures from Gravitational Perturbations

We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel to and return. We apply the result to two concrete examples when the source fields are scalar fields (i.e., Einstein–scalar field equations) and electromagnetic fields (i.e., Einstein-Maxwell equations). © 2019 Wiley Periodicals, Inc.     

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

Real hypersurfaces of a complex projective space

In this paper, we classify real hypersurfaces in the complex projective space C P^\fracn+12C P^{\frac{n+1}{2}} whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.     

Spontaneous particle creation within the external field approximation of QED

It is expected that strong electromagnetic fields in QED may lead to destabilization of the vacuum, its decay and spontaneous production of an electron-positron pair if the strength of the electric field exceeds some threshold. We study the problem within an external field approximation of QED in presence of time-dependent external fields and define the spontaneous particle creation via the adiabatic limit. We consider some model fields, which we solve analytically or numerically, in order to explain the subtlety of the effect and to recognize problems appearing in its proof. We conclude by characterizing when the effect can exist in a stable way and when it becomes unstable w.r.t. small perturbations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)