一种产生Einstein约化场方程解的方法

本文给出了在Ernst场方程的解空间上的一种新变换,并且研究了这种变换与Ernst场方程解之间的关系。证明了在这种变换下,Ernst场方程是不变的,即由我们的这种变换可产生Ernst场方程的新解。最后还讨论了这种变换与Virasoro代数的关系。 关键词：     

Lax pair, conserved currents, and hidden symmetry transformation for the Ernst equation

The matrix Ernst equation (a reduced form of the self-dual Yang-Mills equation) is written as the compatibility condition for solution of a linear “inverse scattering” system. This system is used to construct infinite sequences of nonlocal conserved charges, as well as an infinitesimal hidden symmetry transformation, for the Ernst equation.     

Soliton theoretic framework for generating multimonopoles

We present a systematic method using Bäcklund transformation for generating SU(2) Yang-Mills-Higgs monopoles of arbitrary charge. The purely algebraic iteration formula for our Bäcklund transformation is derived. Our method is based on the equivalence of the axially and mirror-symmetric Bogomolny equations and the Ernst equation. The properties of the Ernst equation that are relevant for monopoles are also discussed. the application of the method is illustrated for the example of the one- and two-monopole solutions.     

Bäcklund transformations of the axially symmetric stationary Einstein-Maxwell equations II

We obtained a Bäcklund transformation of the axially symmetric stationary Einstein-Maxwell equation which is a generalization of Neugebauer's I₁ transformation in the case of the Ernst equations. This transformation is shown to generate the generalized Ehlers transformation and the Harrison transformation.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

Solving Bäcklund transformations with harmonic mappings via Killing vectors

On the basis of Killing vectors, a systematic method for solving Bäcklund transformations of the Euler equation in the harmonic mapping theory is presented. As an application, the Ernst equation for stationary axisymmetrical gravitation is discussed, and the Ehler transformation of that equation is obtained.     

Sigma model for Ernst equation: Lax representation,Bäcklund transformation and divergence-free currents

A sigma model associated with the Ernst equation is derived. This sigma model is described by the Belinsky-Zakharov-type completely integrable equation and is formally equivalent to the usual sigma model in curved two-dimensional space. The corresponding Lax representation, BÄcklund transformation, and divergence-free currents are obtained.     

Double Bäcklund transformations and fission solutions of axisymmetric gravitational equations

The results of Kyriakopoulos are extended to a double-complex form, and a double Bäcklund transformation of the Ernst equation is derived concretely. By using the noncommutative relation between this double Bäcklund transformation and the dual mapping, a fission-type generation process of new solutions is discussed.     

Proof of a Generalized Geroch Conjecture for the Hyperbolic Ernst Equation

We enunciate and prove here a generalization of Geroch's famous conjecture concerning analytic solutions of the elliptic Ernst equation. Our generalization is stated for solutions of the hyperbolic Ernst equation that are not necessarily analytic, although it can be formulated also for solutions of the elliptic Ernst equation that are nowhere axis-accessible.     

关于稳态轴对称真空引力场方程的两个扩展解

将Ehlers变换应用于Ernst方程的Schwarzschild解和Kerr解，通过引入Boyer-Lindquist坐 标变换以及相关的参数代换，得到了Ernst方程的两个扩展解. 当所含参数L=0时，其中一个扩展解退化为Schwarzschild解，另一个退化为Kerr解.当参数|L|M时，如果取近似1-LM2≈1，则这两个扩展解分别退化为已知的NUT-Taub解和Kerr-NUT解.这一结果表明 NUT-Taub解和Kerr-NUT解中所含的参数l并非能任意取值，它的取值要受到引力源质量M的限 制，即要求|l|M. 关键词： Ehlers变换群 Ernst方程 Boyer-Lindquist坐标变换     

SELF-DUAL YANG-MILLS FIELDS IN STATIC AXIALLY SYMMETRIC CASE AND RELATED TOPICS

In this article we will present an explicit geometric picture about the complete integrability of the static axially symmetric SDYM equation and the gravitational Ernst equation, interpret the correspondence between their Bäcklund transformation formulae and the transformations from one focal surface of Weingarten congruence to the integrability of the B.T.will be proved. It is shown that for the axially symmetric SDYM equation and gravitational Ernst equation the adjoint space of the group (SL(2r)) is a 3-dimensional Minkowski space, and the corresponding soliton surfaces have negative variable curvature. After introducing the generator R we can explain the B.T. as the rotation around the common tangent between two surfaces of solitons. Using Roccato eqiatopm we will confirm in this paper the integrability of B.T. amd prpve that the B.T. is strong, i. e. , the nwe and old solutions satisfy equations of motion separately. Some related topics are also discussed.     

VIRASORO ALGEBRA FOR THE THEORY OF THE REDUCED GRAVITY

In the paper we give a new transformation in the reduced gravity, which is associated with the Virasoro algebra without the center term. To compare with the Geroch group it is easy to establish the relationship between the Virasoro algebra and the Kac-Moody algebra in the theory of the reduced gravity. Then we point out that the transformation can be adopted to generated the new solutions of the Ernst equation from the old ones.     

The Ernst equation as a motion on a universal Grassmann manifold

We find a linearization of the Ernst equation by means of universal Grassmann manifold (UGM) techniques. All local analytic solutions defined at the origin are obtained by solving an initial value problem for a linear differential equation on a UGM. We give an explicit formula which represents solutions of the Ernst equation. By using this formula, we generate several special solutions.     

On a discretization of asymptotic nets

We discretize some notions of the theory of asymptotic nets and of the theory of transformations of asymptotic nets. These are the Lelieuvre formulas, the Moutard equation, the Moutard transformation, the Weingarten congruences and the Jonas formulas. It allows us to extend the theory of reductions of the discrete version of the Darboux system, applied primarily to multidimensional quadrilateral lattices, on the theory of discrete asymptotic nets which in turn is helpful in a discretization of some classical differential nonlinear integrable systems of physical interest, e.g. the Ernst equation and the stationary modified Nizhnik–Veselov–Novikov system (in form which we call the Fubini–Ragazzi system).     

Bogomolny Field Equations and the Double-Complex Function Method

The double-complex function method is applied tothe Bogomolny field equations ofYang–Mills–Higgs theory. We discuss how togenerate a family of solutions of the Bogomolny fieldequations with a double-complex Ernst potential. We discuss the application ofthe Neugebauer–Kramer transformation, the doubleEhlers transformation, and noncommutativity. An exampleof a concrete calculation is given.     

Three-dimensional representation of the double quantum algebrasu q(η(J)) and theq-deformation of the double complex ernst equation

A three-dimensional representation of the double quantum algebrasu _q((J)) is given. By the use of this representation and a Lax pair, we obtain a nonlinear Ernst equation system. By the harmonic function method, a solution of theq-deformed double complex Ernst equation is given.     

Multiple Lax pair for the double Ernst equation

The double-complex function method is used to find a multiple Lax pair for the double Ernst equation. Then multiple hidden symmetries and multiple infinite sequences of nonlocal conserved charges are obtained. The result of Papachristou and Harrison is contained in our result.     

Double harmonic mappings of Riemannian manifolds and its applications to stationary axisymmetric gravitational fields

Harmonic mappings of Riemannian manifolds are discussed by a double complex function method, and the double-complex Ernst equation and the related Bäcklund transformations are naturally derived. Further, the Ernst solution and its dual solution are obtained by two different methods, respectively. Therefore, the results obtained by A. Eris are extended to a double form.     

调和映射理论中Euler方程的解和它在广义相对论中的应用

本文讨论了调和映射理论中Euler方程的解以及解与解之间的关系。并以此观点出发,研究了Ernst方程和该方程的解。 关键词：     

Some remarks on finite-gap solutions of the Ernst equation

The relationship between the class of algebro-geometrical (finite-gap) solutions of the Ernst equation constructed by Korotkin and Matveev [Theor. Math. Phys. 77 (1989) 1018; St. Petersburg Math. J. 1 (1990) 379] and the solutions recently constructed by Meinel and Neugebauer [Phys. Lett. A 210 (1996) 160] is discussed. A new formula for the general algebro-geometrical solution is obtained.