Double Complex Function Method Research Articles

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

Abstract We present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.

DOUBLE STRUCTURES AND DOUBLE SYMMETRIES FOR THE EINSTEIN–KALB–RAMOND THEORY

We further study the two-dimensional reduced Einstein–Kalb–Ramond (EKR) theory in the axisymmetric case by using the so-called double-complex function method. We find a doubleness symmetry of this theory and exploit it so that some double-complex d×d matrix Ernst-like potential can be constructed, and the associated equations of motion can be extended into a double-complex matrix Ernst-like form. Then we give a double symmetry group [Formula: see text] for the EKR theory and verify that its action can be realized concisely by a double-complex matrix, form generalization of the fractional linear transformation on the Ernst potential. These results demonstrate that the theory under consideration possesses more and richer symmetry structures. Moreover, as an application, we obtain an infinite chain of double-solutions of the EKR theory showing that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.

Double structures and double symmetries for the general symplectic gravity models

By using the so-called double-complex function method, a doubleness symmetry for each member of the class of stationary axisymmetric general symplectic gravity models is found and exploited so that some double-complex (n+1)×(n+1) matrix Ernst-like potential for any non-negative integer n can be constructed and the associated motion equations can be extended into a double-complex matrix Ernst-like form. Then double symmetry symplectic groups Sp(2(n+1), R(J)) of the theories are given and verified that their actions can be realized concisely by double-complex matrix form generalizations of the fractional linear transformation on the Ernst potential. These results demonstrate that the theories under consideration possess more and richer symmetry structures. The special cases n=0 and n=1 correspond, respectively, to the pure Einstein gravity and the Einstein–Maxwell-dilaton–axion theories. Moreover, as an application, for each n=0,1,2,…, an infinite chain of double-solutions of the general symplectic gravity model is obtained, which shows that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.

Double Structures and Double Symmetries for the Einstein-Maxwell-Dilaton-Axion Theory

By using the so-called double-complex function method, it is found that the stationary axisymmetric D = 4 Einstein-Maxwell-Dilaton-Axion system can be written in a double-complex matrix Ernst-like form. Then the double symmetry symplectic group Sp((4, R(J)) of the theory and its double-complex fractional linear realization are given. These results demonstrate that the theory considered possesses more and richer symmetry structures than previously expected. Moreover, as an application, an infinite chain of double-solutions of the problem is obtained, which shows that the double-complex method is more effective. Some of the results of this paper cannot be obtained by the usual (non-double) scheme.

The Double Self-Dual Gravity with Cosmological Term and Constraints Under the Double Constant Conformal Transformation**The project supported by Colleges Science Research Foundation of Educational Department of Liaoning Province of China under Grant Nos 20041012 and 990111004, and Natural Science Foundation of Liaoning Provence of China under Grant No. 002083

The double constraint equations in the self-dual gravitational theory containing the cosmological term are derived in gravity. Furthermore, in order to deeply study the Lorentzian and Euclidean reality conditions for this theory, the relations between constraints are discussed by introducing the double constant conformal transformation and the double complex function method.

On the problem of reality conditions for the Ashtekar formulation

In this paper we discuss the reality conditions for Lorentzian and Euclidean gravity in the Ashtekar formulation by introducing a double conformal transformation. We generalize Marugan's results and demonstrate that the values of the double conformal factor have to be either real or double complex numbers. Either Lorentzian or Euclidean gravitational theory is up to the different values of the double conformal factor. Furthermore, the reality conditions of Lorentzian and Euclidean gravitational theory can be expressed in a unified way by use of the double complex function method.

New ways of deriving Arnowitt-Deser-Misner constraint equations in four-dimensional gravity

In this paper, the Arnowitt-Deser-Misner (ADM) constraint equations are naturally derived in two different ways. One method is to construct a one-parametric gravitational action in the Lorentzian spacetime. Hence, the one-parametric ADM constraint equations can be obtained. The other method is to apply the double complex function method to Einstein-Hilbert gravitational fields in Hamiltonian formulation. Therefore the double ADM constraint equations can be obtained, in which the well-known ADM constraint equations are included as a special case.

Double Complex (Anti-)self-dual Variables and a Related Double Action in Four-dimensional Gravity

In this paper double dual transformations anddouble complex (anti-)self-dual variables infour-dimensional gravity are given by using the doublecomplex function method. A double action with doublecomplex anti-self-dual variables is proposed, hence thedouble complex (anti-) self-dual gravitational theory isestablish. Finally, the analytic continuation from thedouble action to the Euclidean action is discussed, and some given results are here extended intoa double form.

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.

PARAMETRIC (ANTI-) SELF-DUAL VARIABLES AND A RELATED PARAMETRIC YANG-MILLS-LIKE ACTION IN FOUR-DIMENSIONAL GRAVITY

Internal and external parametric dualtransformations as well as (anti-) self-dual variablesin four-dimensional gravity are presented in a unifiedway via a parameter. The double complex (anti-)self-dual variables including the double complex Ashtekarvariables are studied by using the double complexfunction method. Concretely, a parametricYang–Mills-like action is proposed and theanalytic continuation from a double action to the Euclidean action isdiscussed. Some results of previous gravitationaltheories are extended to a parametric form.

Extended double soliton solution families for the statically axisymmetric self-dualSU(2) gauge field equations

By using the extended double complex function method, the statically axisymmetric self-dualSU(2) gauge field equations and, in turn, the Belinsky-Zakharov solution-generating technique are generalized to extended double forms. The restriction on “soliton index” in the original solution-generating technique is eliminated so that for each positive integer, we can obtainphysical soliton solutions of the statically axisymmetric self-dualSU(2) gauge field equations in pairs. Some sufficient conditions are given for seed solutions with which the corresponding scattering wave function can be written out directly. As examples, some soliton solution families are given, most solutions of which are new.

Extended double Riemann–Hilbert problem and multiple representations of the infinite-dimensional symmetric group for the stationary axisymmetric vacuum field equations

The double-complex function method suggested in previous papers is extended and the extended method is used to establish two mutually dual extended double Hauser–Ernst equations and related Riemann–Hilbert problems, then quadruple representations of the semidirect product of the Kac–Moody and Virasoro groups are given. These show that, for the stationary axisymmetric vacuum space–times, there are more and richer symmetry structures than previously expected. The multiple forms of some well-known transformations can be obtained as some special cases.

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 Backlund 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.

Regular double Riemann–Hilbert problems and double Kac–Moody algebraic structures for two-dimensional reduced gravity

By using the double-complex function method, two regular double Riemann–Hilbert problems are established; therefore two double Kac–Moody algebras are given. These structures show that in two-dimensional reduced gravity, in fact, there is more exquisite hidden symmetry than common nonlinear systems.

Generation of new solutions of the stationary axisymmetric Einstein equations by a double complex function method

A new approach to the solution of certain differential equations, the double complex function method, is developed, combining ordinary complex numbers and hyperbolic complex numbers. This method is applied to the theory of stationary axisymmetric Einstein equations in general relativity. A family of exact double solutions, double transformation groups, and n-soliton double solutions are obtained.