Generating conjecture and Einstein-Maxwel field of plane symmetry

For the plane symmetry we have found the electro-vacuum exact solutions of the Einstein-Maxwell equations and we have shown that one of them is equivalent to the McVittie solution of a charged infinite thin plane. The analytical extension has been accomplished and the Penrose conformal diagram has been obtained as well.     

Some type III solutions of the Einstein-Maxwell equations

A Petrov type III metric with nontwisting, degenerate Debever-Penrose direction is studied. This metric is, in general, a solution of the Einstein-Maxwell equations. Two particular cases are investigated in some detail. It is shown that the metric contains typeN, conformally flat and flat metrics as special subcases. Among these subcases, we find the metric of plane gravitational waves and the Bertotti-Robinson solution.     

Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schrödinger Equation

Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multi-symplectic splitting(MSS) method to solve the two-dimensional nonlinear Schrödinger equation (2D-NLSE) in this paper. It is further shown that the method constructed in this way preserve the global symplecticity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness
of the proposed method.     

A new exact solution for colliding gravitational plane waves

A new exact solution of the vacuum Einstein equations describing the spacetime following the collision of two plane impulsive gravitational waves, each supporting a plane gravitational wave, is obtained. The solution has been extended prior to the instant of collision and the main features of the resulting space-time have been analyzed, using the Newman-Penrose formalism. It is shown that the result of the collision is the development of a singularity of the spacetime due to the simultaneous focussing of the two plane waves.     

Equivalence of self-consistency and conservation of energy in the scattering of an electromagnetic wave by a plane of dipoles

The scattering of a harmonic electromagnetic plane wave by a plane of induced dipoles has been rigorously treated by means of the Hertz vector. We have shown that only a self-consistent solution, which takes into account cooperative effects among scatterers, is able to conserve energy.     

The spin-density waves in thin films of Cr in the Fe/Cr/Sn/Cr multilayers

Magnetism of Fe/Cr/Sn/Cr multilayers have been studied by the first-principles density-functional theory. Calculations by the full-potential linearized augmented plane waves (FP LAPW) and screened Korringa–Kohn–Rostoker (SKKR) methods have shown that two types of solutions may be found in these systems: a high-spin (HS) and a low-spin (LS) ones in accordance with the Cr magnetic moment magnitudes. In this work, we have concentrated our attention on the LS solution. The calculation has shown that the LS solution has a phase slip of antiferromagnetic ordering in Cr and is classified as an incommensurate spin density wave (ISDW). The dependence of the ISDW solution on the thickness of Cr, lattice parameter and boundary conditions has been investigated.     

非线性声学谐波方程的特解及其在边值问题中的应用

在流体或固体介质中,微扰法求解非线性声波的反射和折射问题时,谐波场通常满足非齐次波动方程。应用分离变量法及拉格朗日变动参数法求它的特解,出现了待定的分离常数,给这类非线性声学边值问题带来困难。本文结果表明,为不使定解问题出现佯僇,其独立的特解只有两类,一类是沿边界法线方向有积累效应的解,另一类则是沿平行边界面方向上有积累效应的解,由定解条件来决定究竟选用哪一类解。应用这个结果研究了平面边界的反射和折射谐波,该理论对非线性声学中的平面边值问题有普遍的应用意义。     

Modal theory for the two-frequency mutual coherence function in random media: point source

The recently introduced modal expansion representation for the two-frequency mutual coherence function is applied here to the solution of a point-source field in a random medium. This approach reduces the solution for any structure function to an eigenvalue problem for an ordinary differential equation. For the initial point source it is shown here that the modal expansion yields a result similar to that for the initial plane wave, modified by a spherical free-space phase which contains a weighted coordinate that does not interact with the medium. Having established these general characteristics, special attention is paid to power-law media and, in particular, to a quadratic medium, for which a new exact solution is derived. Via a collective summation of this new modal solution, we rederive the alternative exact solution which exists in the literature. We also discuss the new parameterization implied by the new modal solution.     

Diffraction from a half-plane. A new derivation of the Sommerfeld solution

It is shown that the Sommerfeld solution for the diffraction of a plane wave by a perfectly conducting half-plane can be obtained in an elementary manner by a superposition of anisotropic cylindrical waves.     

A non-static source of the Taub solution of Einstein's gravitational equations

It is shown that a collapsing (anticollapsing) plane desk consisting of incoherent dust can be the source of the Taub solution. The metric tensor inside the matter is found.     

Field generated by an anisotropic conducting plane in a half-space with chiral properties

This article inaugurates theoretical studies on a new class of composite structures that are a combination of anisotropically conducting surfaces and media possessing chiral properties. To test the mathematics, we solved the simplest model of the problem in which an in-phase electric current filament excites a plane formed by a layer of straight conductors at the interface of two isotropic media, one of which is chiral. The solution is given as a Fourier integral. As shown by analysis, one retarded wave of a discrete spectrum and three waves of a continuous spectrum are excited in the given structure. The first one exists only because of the anisotropically conducting plane and does not exist without it, while the latter three are associated with plane waves of corresponding homogeneous media, magnetodielectric and chiral.V. D. Kuznetsov Siberian Physicotechnical Institute, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 34–38, September, 1994.     

The gravitational field of a homogeneous plate with a non-zero cosmological constant

A static interior solution of Einstein gravitational equations with λ ≠ 0 generated by a plane plate of a constant mass density is found. It is shown how to join this interior solution with the exterior metric as well     

Three-dimensional analysis of scattering by pressure-release plane surfaces and the validity of the image solution

Because of the complexity of the scattering integrals in three dimensions, numerous approximations are used to obtain closed-form solutions. By considering the scattering by an infinite, pressure-release plane surface, the effects of various phase approximations and source directivity approximations can be examined independently of the surface roughness. Calculations are carried out using the Fraunhofer and Fresnel phase approximations, and two directivity approximations. It has been shown experimentally that the image solution is valid for the reflection of an acoustic beam by an infinite, pressure-release plane surface if the plane is in the farfield of the source. Consequently, the image solution is used to compare analytical solutions obtained using various phase and directivity approximations, and it is found that both the Fresnel phase approximation and a realistic directivity approximation are required to achieve a good fit. The solution produced by the Fraunhofer phase approximation is obtained as an asymptotic limit of the modified Fresnel solution. Criteria for the validity of the Fraunhofer and Fresnel phase approximations are developed. The Fresnel phase approximation is valid under fairly broad conditions, but the Fraunhofer phase approximation is never valid for an infinite plane surface that must be in the farfield of the source.     

Resonant light scattering by a gradient quantum well

Resonant scattering of monochromatic light by a quantum well with the frequency of its excitonic resonance varying along a certain direction in the plane of the well is studied experimentally and theoretically. It is shown that the simplest model of a thin inhomogeneous layer that yields an exact solution of the direct and inverse scattering problems allows one to successfully describe the experimental observation of resonant scattering by a GaAs/AlGaAs quantum well with the frequency of the exciton resonance linearly varying in the plane of the well.     

Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet

An analytical study on the properties of intrinsic localized modes and modulational instability in a quantum two-dimensional ferromagnet with single-ion uniaxial anisotropy is completed in the semiclassical limit. By making use of the semidiscrete multiple-scale method, we obtain a line localized solution and a radially symmetric localized solution, and analyze their existence conditions. Taking into account that the existence of bright localized solutions is closely linked to the modulational instability of plane waves, we analytically study the discrete modulational instability of plane spin waves. The result of the modulational instability analysis show that the uniaxial anisotropy plays a key role in the appearance of our intrinsic localized spin wave modes.     

Supernonlinear ion-acoustic waves in a dusty plasma

An exact solution is obtained for the equations that describe nonlinear ion-acoustic waves in a dusty plasma. It is shown that the solution can be in the form of nonlinear periodic waves, solitons, and supernonlinear waves whose trajectories envelope one or several separatrices in the phase portrait of the wave. Profiles of physical quantities in the wave are constructed. The supernonlinear waves are shown to be of two types, subsonic (type 1) and supersonic (type 2). Existence regions of supernonlinear waves of both types and solitons are constructed in the plane of the problem parameters.     

Diffraction of electromagnetic plane wave by a slit in a homogeneous bi-isotropic medium

We investigated the diffraction of an electromagnetic plane wave by an infinite slit embedded in a homogeneous bi-isotropic medium. With the aim of deriving explicit expressions for the left- and right-handed Beltrami fields, we used the Fourier integral transform, the Wiener–Hopf technique and the steepest descent asymptotic method. The electric and magnetic fields, E and H, were determined from the Beltrami fields. Our graphical results indicate that the strength of both electric and magnetic fields reduces with the dissipation of bi-isotropic medium. While matching the diffraction pattern with the existing plane wave solution, the objective was, and is, to see how well spherical wave solution performs when it is developed for plane wave solution.     

Integrability of auto-Bäcklund transformations and solutions of a torqued ABS equation

An auto-Bäcklund transformation for the quad equation Q1₁ is considered as a discrete equation, called H2^a, which is a so called torqued version of H2. The equations H2^a and Q1₁ compose a consistent cube, from which an auto-Bäcklund transformation and a Lax pair for H2^a are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund transformation for H2^a we derive a seed solution and a one-soliton solution. From this solution it is seen that H2^a is a semi-autonomous lattice equation, as the spacing parameter q depends on m but it disappears from the plane wave factor.     

On the gravitational field of a plane plate in general relativity

It is shown that a static solution of the Einstein equations inside an infinite plate of an ideal liquid with continuous metric coefficients and their first derivatives cannot have a plane of mirror symmetry. As a consequence, the boundaries of the plate are joined with qualitatively different vacuum solutions on both sides of the plate.     

Interaction of plane electromagnetic-gravitational waves

The solution in the general theory of relativity of the problem of the interaction of plane electromagnetic-gravitational waves is derived in this paper by means of an electromagnetic generalization of the problem of the collision of plane gravitational pulses. It is shown that the interaction of waves leads to a physical singularity on a spacelike hypersurface.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 65–70, October, 1978.In conclusion, the author thanks Professor D. D. Ivanenko for his comments.