非线性波方程求解的新方法

从Legendre椭圆积分和Jacobi椭圆函数的定义出发，得到了新的变换，并把它用于非线性演化方程的求解.用三个具体的例子，如非线性Klein-Gordon方程、Boussinesq方程和耦合的mKdV方程组，说明了具体的求解步骤.比较方便地得到非线性演化方程或方程组的新解析解，如周期解、孤子解等. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤子解     

Exact solutions to Einstein field equations

In a recent paper Reboucas and d'Olival obtain an ordinary differential equation for a Bianchi type II metric with a rotating timelike congruence of geodesics, and obtain a particular solution of the differential equation. This paper completely integrates the differential equation.     

On solutions of Einstein and Einstein-Yang-Mills equations with (maximal) conformal subsymmetries

We consider the maximal subgroups of the conformai group (which have in common as a subgroup the group of pure spatial rotations) as isometry groups of conformally flat spacetimes. We identify the corresponding cosmological solutions of Einstein's field equations. For each of them, we investigate the possibility that it could be generated by anSU (2) Yang-Mills field built, via the Corrigan-Fairlie-'t Hooft-Wilczek ansatz, from a scalar field identical with the square root of the conformal factor defining the space-time metric tensor. In particular, the Einstein cosmological model can be generated in this manner, but in the framework of strong gravity only, a micro-Einstein universe being then viewed as a possible model for a hadron.Boursier A.G.C.D.     

On Chelnokov-Zeitlin solutions to the vacuum Einstein equations

The equivalence of the Chelnokov-Zeitlin solutions to the vacuum Einstein equations with a special class of Lewis solutions is established in a direct way. Also, an oversight on the signature of the solutions is pointed out and corrected.     

New stiff matter solutions to Einstein equations

New exact solutions are presented to the Einstein field equations which are spherically symmetric and static, with a perfect fluid distribution of matter satisfying the equation of state=p. One of the obtained solutions may only be used locally, the other represents the stellar interior globally and is singularity-free.     

Approximate radiative solutions of the Einstein equations

In this paper the external field of a bounded source emitting gravitational radiation has been considered. A successive approximation method has been used to integrate the Einstein equations in Bondi's coordinates. A method of separation of angular variables has been worked out and the approximate Einstein equations have been reduced to the key equations (3.8)–(3.10). The losses of mass, momentum, and angular momentum due to gravitational multipole radiation have been found. It has been demonstrated that in the case of proper treatment a real mass occurs instead of a mass aspect in a solution of the Einstein equations. In Appendix C Bondi's news function has been given in terms of sources.     

Exact vacuum solutions of the Einstein equations

The vacuum Einstein equations for the Kerr-Schild metric are investigated. It is shown that they admit representation in the form of the double four-dimensional curl of the perturbation of the Euclidean metric, whereupon it is possible to note certain general directions in which to seek exact solutions. For spaces with a normal isotropic geodesic congruence the GR equations are rewritten with the application of a dyadic splitting of the metric; cases of two-dimensional subspaces of constant curvature are discussed. The investigation is illustrated by the exact nonstationary algebraic type N and anti-Schwarzschild solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 23–26, September, 1982.     

构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

The method of conformal mapping and new solutions of the einstein equations with the energy-momentum tensor of an ideal fluid

Two solutions with the energy-momentum tensor of an ideal fluid are obtained on the basis of theorems proved earlier by the author on the possibility of constructing new solutions of the Einstein equations from solutions already known by means of conformal mapping. The first of these solutions is a conformal correspondent to the De Sitter solution, and the second corresponds to the class of Friedman solutions. The explicit form of the metrics of the new solutions and of the parameters of the energy-momentum tensor is written out, and the properties of the corresponding space-times are also investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 111–115, April, 1977.     

Space-time defect solutions of the Einstein field equations

Ideas from the theory of defects in crystalline matter are combined with results from the direct gauge theory for the Poincaré group to obtain exact solutions of the Einstein field equations. Many of the solutions are sufficiently simple that the equations for geodesic motion can be solved in closed form. Some of these solutions exhibit unexpected behaviors and properties, such as geodesic motions with hyperlight speed and local time reversals relative to observers in the asymptotic Minkowski space-time at large distances from the defect core regions. However, these same geodesic motions are regular in the frames of reference attached to observers that move along the geodesies, and hence no established physical laws are broken by such solutions.     

On certain new exact solutions of the Einstein equations for axisymmetric rotating fields

We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations （PDEs） and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.     

Exact solutions of vacuum n-dimensional Einstein equations

A class of exact solutions of the vacuum n-dimensional Einstein equations for the case in which the components of the metric tensor depend on two variables is obtained by the method of separation of variables. Particular cases of this class of solutions are considered: a plane-symmetric metric; an axisymmetric metric; metrics of the Casner type.M. V. Lomonosov State University, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 120–124, October, 1995.     

Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

Semi-global solutions of Einstein equations,minkowskian near past infinity

On a universe homeomorphic toV ^T =]– ,T[x³, we prove the existence of solutions of Einstein equations, minkowskian near past infinity, if the sources are small enough for some norms. We prove that some of these solutions verify at least the positivity condition (Weak energy condition) on some domains homeomorphic toV ^T .     

Republication of: A new class of vacuum solutions of the Einstein field equations

This reprinting of a paper by Kerr and Schild, first published in 1965 in a conference volume that is difficult to get hold of today, has been selected by the Editors of General Relativity and Gravitation for publication in the Golden Oldies series of the journal. It is the only publication showing how the Kerr solution was originally arrived at. In this reprint the editors have added several footnotes that update the references to the literature. The paper is accompanied by an editorial note written by A. Krasiński, E.Verdaguer and R. P. Kerr, and by the biography of Kerr written by A. Krasiński.     

Discontinuities of solutions of the Einstein equations. II

Right zero-vectors of the characteristic matrix of the Einstein equations are constructed on isotropic cones. Relationships for the discontinuities of two derived functions of the field in the and surfaces are indicated. Quantities describing the weak discontinuities of solutions of the gravitational field equations are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–41, February, 1982.