The electromagnetic initial boundary value problem in a domain with a chiral filling

The problem of the excitation of electromagnetic oscillations by given charge and current distributions in a domain with a nonhomogeneous chiral filling is investigated. The domain where the problem is considered may be both a finite one bounded by an ideally conducting surface and an infinite supplement to an ideally conducting bounded object. The initial boundary value problem is shown to arise, for which the generalized formulation in a special functional space is given. The existence of a unique weak solution to the problem is proven using the Galerkin method.     

对称圆环状多原子分子的自由振动

考虑对称的圆环状多原子分子的振动问题,给出了该振动系统简正频率的数学表达式和自由振动的一般解.     

Fractional diffusion equation in half-space with Robin boundary condition

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.     

Ordering energy and stability of perfect superlattices

The rigorous solution to the problem of stability of stochiometric ordered phases is considered. Conditions of stability are found relative to phases with maximal degree of order for given stochiometry. The treatment is given for all known superlattices that arise from an fcc solid solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 22–26, September, 1980.     

Stochastic radiative transfer in a finite plane with anisotropic scattering in a binary Markovian mixture

The time-independent linear transport problem in a stochastic finite-plane medium with linear anisotropic scattering is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The Pomraning-Eddington approach is used to obtain an explicit solution to the problem in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution for the problem under consideration. In the case of isotropic scattering, explicit analytic results for reflectivity and transmissivity, which show a good agreement with Monte Carlo benchmark results, are given. Results for reflectivity and transmissivity in the case of linear anisotropic scattering are also given.     

On the Frenkel problem of equivalent steady currents in a sphere

The Frenkel problem of substituting the 3D system of steady currents given in one of two concentric spherical regions by an equivalent system of currents (i.e., by that inducing the same external magnetic field) that is distributed over the surface of the other region is considered. A method of multipole moments providing the direct solution (without calculating the fields) of the problem is described. The case of currents with the density components represented by cubic polynomials of the Cartesian coordinates is considered as an example.     

Discretization and its proof for numerical solution of a Cauchy problem for Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface

Cauchy problem for the Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface is considered. Discretization was performed and proved to obtain a numerical solution. An economic algorithm is proposed. The text was submitted by the authors in English.     

Analytical solution of a system of magnetohydrodynamic equations in a quasi-one-dimensional approximation for the monotonic variation of the flow velocity along a channel

An analytical solution is obtained for the system of magnetohydrodynamic (MHD) equations in a quasi-one-dimensional approximation for flow regimes with a monotonic variation of the velocity along a MHD channel. The problem of optimal choice of the parameters of an MHD generator for the given ratio of the areas of the outlet and inlet MHD channel cross sections is considered using the resultant analytic solution.     

Soliton dynamics in the Wess-Zumino-Novikov-Witten model

Exact solutions for the classical Wess-Zumino-Novikov-Witten model are constructed by the matrix Darboux transformation method. The behavior of one- and two-soliton solutions is analyzed. The formula for an N-soliton solution is given. The corresponding linearized problem is considered.     

Knudsen Layer for Gas Mixtures

The Knudsen layer in rarefied gas dynamics is essentially described by a half-space boundary-value problem of the linearized Boltzmann equation, in which the incoming data are specified on the boundary and the solution is assumed to be bounded at infinity (Milne problem). This problem is considered for a binary mixture of hard-sphere gases, and the existence and uniqueness of the solution, as well as some asymptotic properties, are proved. The proof is an extension of that of the corresponding theorem for a single-component gas given by Bardos, Caflisch, and Nicolaenko [Comm. Pure Appl. Math. 39:323 (1986)]. Some estimates on the convergence of the solution in a finite slab to the solution of the Milne problem are also obtained.     

Double-deck structure of the boundary layer in the problem of flow in an axially symmetric pipe with small irregularities on the wall for large Reynolds numbers

The problem of flow of a viscous incompressible fluid in an axially symmetric pipe with small irregularities on the wall is considered. An asymptotic solution of the problem with the double-deck structure of the boundary layer and the unperturbed flow in the environment (the “core flow”) is obtained. The results of flow numerical simulation in the thin and “thick” boundary layers are given.     

Identification of low-energy light nuclei recorded by means of an ionizational mass spectrometer

A method of identifying low-energy nuclei from the readings of an ionizational mass spectrometer is considered. The identification principles for particles recorded by a multilayer detector are discussed, together with the fundamentals of a probabilistic approach to the solution of this problem. The method permits correct analysis of the experimental data obtained. The determination of the charge and isotopic composition of radiation consisting of a mixture of different nuclei is considered. The results of calculations by the given method which permit the determination of the optimal detector parameters for the solution of the specific physical problem are outlined. This approach is also applicable for the analysis of data from other types of measuring apparatus.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 84–88, July, 1991.     

三角形折射率剖面光纤一级修正解的数值计算

在弱波导近似下,用标量Helmholtz方程的解作为零级近似解,考虑了折射率梯度的影响,把∇ε项看作微扰项,用微扰法给出了具有色散位移特性的三角形折射率剖面单模光纤一级修正解的积分表达式,并给出了数值计算结果.原则上,该方法亦可用于多模光纤.     

Solution methods for discrete-state Markovian initial value problems

Solution methods, both numerical and analytical, are considered for solving the Liouville master equation associated with discrete-state Markovian initial value problems. The numerical method, basically a moment (Galerkin) method, is very general and is validated and shown to converge rapidly by comparison with an earlier reported analytical result for the ensemble-averaged transmission of photons through a purely scattering statistical rod. An application of the numerical method to a simple problem in the extended kinetic theory of gases is given. It is also shown that for a certain restricted class of problems, the master equation can be solved analytically using standard Laplace transform techniques. This solution generalizes the analytical solution for the photon transmission problem to a wider class of statistical problems.     

Interacting electromagnetic waves in general relativity

The problem is considered of finding exact solutions of the Einstein-Maxwell equations which describe the physical situation of two colliding and subsequently interacting electromagnetic waves. The general theory of relativity predicts a nonlinear interaction between electromagnetic waves. The situation is described using an approximate geometrical method, and a new exact solution describing two interacting electromagnetic waves is given. This describes waves emitted from two sources mutually focusing each other on the opposite source.     

具有一般非线性弹性力和广义阻尼力的相对转动非线性系统的周期解问题

讨论了一类相对转动非线性动力系统的周期解问题.首先建立了一类具有一般非线性弹性力、广义阻尼力和强迫周期力项的相对转动非线性动力系统;其次得到了对应自治系统的周期解不存在性结果,以及运用Mawhin重合度理论得到了该模型的周期解存在性结果,推广了已有的结果;最后举例证明本文结果的正确性.     

Electric field and the charge distribution on the surface of an insulator in a vacuum

The free charge steady-state distribution over the insulator surface that arises in a strong electric field in a vacuum can be found by solving the boundary-value problem for the electrostatic field strength if the angle between the field vector and vacuum-insulator interface is given. A general solution to this boundary-value problem is derived for the case of an in-plane field and rectilinear interfaces. Laws of charge and field formation that follow from the solution obtained are considered. Formulas for the electric field strength and charge density in terms of elementary functions are obtained for a number of particular cases. Power-type expressions for the electric field and a critical angle between the electrode and insulator surface that describe the field behavior and charge distribution near the vacuum-insulator-electrode contact are derived.     

Electrostatic problem for a planar eccentric ring

The electrostatics problem for a concentric annular disk has been studied fairly comprehensively (see Ref. 1 and the literature cited). An eccentric ring is considered for the first time using an assumption known in the literature as the Kelvin transformation. This transformation allowed the initial problem to be reduced to that for a concentric ring with a point charge lying outside the ring in the same plane. This problem was solved using triple integral equations. An approximate solution for the charge density distribution at an eccentric annular disk is given as an expansion in terms of a small parameter. Zh. Tekh. Fiz. 68, 104–106 (December 1998)     

Mechanical systems with nonlinear constraints

A geometrical formalism for nonlinear nonholonomic Lagrangian systems is developed. The solution of the constrained problem is discussed by using almost product structures along the constraint submanifold. Constrained systems with ideal constraints are also considered, and Chetaev conditions are given in geometrical terms. A Noether theorem is also proved.     

Method for solving the problems of multiple scattering by several bodies in a homogeneous unbounded medium

A method is developed for solving problems of multiple scattering by an aggregate of bodies in a homogeneous unbounded medium. For this purpose, the problem on the multiple scattering produced by two bodies in the field of a plane wave is first considered under the assumption that the initial unperturbed scattering amplitudes of both scatterers are known. The solution is constructed by considering plane waves multiply rescattered by the scatterers. Integral equations are obtained that allow one to calculate the resulting scattering amplitude of each scatterer and the combined scattering amplitude of the system of two scatterers. It is shown that knowledge of the solution to this problem is sufficient to solve the problem on the scattering field of a system consisting of an arbitrary number of scatterers. Expressions for the scattering amplitude in the case of an arbitrary primary field are presented. The relationship between the integral equations describing the multiple scattering in a homogeneous space and the multiple scattering by a single scatterer located near an interface is demonstrated. Approximate expressions are given for calculating the scattering amplitude in the case of multiple scattering.