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The solution generating methods discovered earlier for integrable reductions of Einstein's and Einstein - Maxwell field equations (such as soliton generating techniques, B$\ddot{a}$cklund or symmetry transformations and other group-theoretical methods) can be described explicitly as transformations of especially defined "coordinates" in the infinite-dimensional solution spaces of these equations. In general, the role of such "coordi\-nates", which characterize every local solution, can be performed by the monodromy data of the fundamental solutions of the corresponding spectral problems. However for large subclasses of fields, these can be the values of the Ernst potentials on the boundaries which consist of such degenerate orbits of the space-time isometry group, in which neighbourhood the space-time geometry and electromagnetic fields possess a regular behaviour. In this paper, transformations of such "coordinates", corresponding to different known solution generating procedures are described by simple enough algebraic expressions which do not need any particular choice of the initial (background) solution. Explicit forms of these transformations allow us to find the interrelations between the sets of free parameters, which arise in different solution generating procedures, as well as to determine some physical and geometrical properties of each generating solution even before a detail calculations of all its components., Comment: LaTeX, 41 pages

It is known that two Reissner-Nordstrom black holes or two overextreme Reissner-Nordstrom sources cannot be in physical equilibrium. In the static case such equilibrium is possible only if one of the sources is a black hole and another one is a naked singularity. We define the notion of physical equilibrium in general (stationary) case when both components of a binary system are rotating and show that such system containing a Kerr-Newman black hole and a Kerr-Newman naked singularity also can stay in physical equilibrium. The similar question about the system of two charged rotating black holes or two rotating overextreme charged sources still remains open., Comment: 20 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1211.3964

An exact analytical model of the process of collision and nonlinear interaction of gravitational and/or electromagnetic soliton wave and strong non-soliton electromagnetic traveling wave of arbitrary profile propagating in the expanding universe (the symmetric Kasner spacetime) is presented. In contrast to intuitive expectations that rather strong traveling waves can destroy the soliton, it occurs that the soliton survives during its interaction with electromagnetic wave of arbitrary amplitude and profile, but its parameters begin to evolve under the influence of this interaction. If a traveling electromagnetic wave possesses a finite duration, the soliton parameters after interaction take constant values again, but these values in general are different from those before the interaction. Based on exact solutions of Einstein - Maxwell equations, our model demonstrates a series of nonlinear phenomena, such as (a) creation of gravitational waves in the collision of two electromagnetic waves, (b) creation of electromagnetic soliton wave in the collision of gravitational soliton with traveling electromagnetic wave, (c) scattering of a part of soliton wave in the direction of propagation of traveling electromagnetic wave, (d) quasiperiodic oscillating character of fields in the wave interaction region and multiple mutual transformations of gravitational and electromagnetic waves in this region. The figures illustrate these features of nonlinear wave interactions in General Relativity., Comment: 5 pages and 2 figures; added new references with comments and the Concluding remarks section

Applications of the monodromy transform approach to construction of exact solutions of electrovacuum Einstein - Maxwell field equations are considered. Examples of new solutions are given., Comment: 2 pages. This is the English translation of the text published in 1988 in Russian

Simple exact solutions presented here describe the universes which spatial geometries are asymptotically homogeneous and isotropic near the initial singularity, but which evolution goes under the influence of primordial magnetic fields. In all these "deformed" Friedmann models (spatially flat, open or closed), the initial magnetic fields are concentrated near some axis of symmetry and their lines are the circles -- the lines of the azimuthal coordinate $\varphi$. Caused by the expansion of the universe, the time-dependence of a magnetic field induces (in accordance with the Faraday law) the emergence of source-free electric fields. In comparison with the Friedmann models, the cosmological expansion goes with acceleration in spatial directions across the magnetic field, and with deceleration along the magnetic lines, so that in flat and open models, in fluid comoving coordinates, the lengths of $\varphi$-circles of large enough radius or for late enough times decrease and vanish for $t\to\infty$. This means that in flat and open models, we have a partial dynamical closure of space-time at large distances from the axis, i.e. from the regions where the electromagnetic fields in our solutions are concentrated. To get simple exact solutions of the Einstein-Maxwell and perfect fluid equations, we assume for the perfect fluid (which supports the isotropic and homogeneous "background" Friedmann geometries) rather exotic, stiff matter equation of state $\varepsilon=p$. However, it seems reasonable to expect that similar effects might take place in the mutual dynamics of geometry and strong electromagnetic fields in the universes with more realistic matter equations of state., Comment: 14 pages, 1 figure

This paper is a direct continuation of our old publication where it was found the exact solution of the Einstein-Maxwell equations for two static sources of Reissner-Nordstrom type in the state of the physical equilibrium. Here we present the exact solution of these equations for the case of two rotating charged sources and we proved the existence of the physical equilibrium state also for this general case.

The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity and supergravity in four and higher dimensions. For different types of fields in space-times of $D\ge 4$ dimensions with $d=D-2$ commuting isometries -- stationary fields with spatial symmetries, interacting waves or partially inhomogeneous cosmological models, the string gravity equations govern the dynamics of interacting gravitational, dilaton, antisymmetric tensor and any number $n\ge 0$ of Abelian vector gauge fields (all depending only on two coordinates). The equivalent spectral problem constructed earlier allows to parameterize the infinite-dimensional space of local solutions of these equations by two pairs of \cal{arbitrary} coordinate-independent holomorphic $d\times d$- and $d\times n$- matrix functions ${\mathbf{u}_\pm(w), \mathbf{v}_\pm(w)}$ of a spectral parameter $w$ which constitute a complete set of monodromy data for normalized fundamental solution of this spectral problem. The "direct" and "inverse" problems of such monodromy transform --- calculating the monodromy data for any local solution and constructing the field configurations for any chosen monodromy data always admit unique solutions. We construct the linear singular integral equations which solve the inverse problem. For any \emph{rational} and \emph{analytically matched} (i.e. $\mathbf{u}_+(w)\equiv\mathbf{u}_-(w)$ and $\mathbf{v}_+(w)\equiv\mathbf{v}_-(w)$) monodromy data the solution for string gravity equations can be found explicitly. Simple reductions of the space of monodromy data leads to the similar constructions for solving of other integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or vacuum gravity in $D\ge 4$ dimensions., Comment: RevTex 7 pages, 1 figure

New derivation of static equilibrium state for two charged masses in General Relativity is given in the framework of the Inverse Scattering Method as an alternative to our previous derivation of this solution by the Integral Equation Method. This shows that such solution is of solitonic character and represents the particular case of more general (12-parametric) stationary axisymmetric electrovacuum two-soliton solution for two rotating charged objects obtained by one of the authors in 1986. This result gives an additional support to our comprehension that the appropriate analytical continuations of solitonic solutions in the space of their parameters are always possible and that applicability of the Inverse Scattering Method in presence of electromagnetic field is not restricted only to the cases with naked singularities., Comment: 7 pages, RevTeX4

More than thirty years passed since the first discoveries of various aspects of integrability of the symmetry reduced vacuum Einstein equations and electrovacuum Einstein - Maxwell equations were made and gave rise to constructions of powerful solution generating methods for these equations. In the subsequent papers, the inverse scattering approach and soliton generating techniques, B\"acklund and symmetry transformations, formulations of auxiliary Riemann-Hilbert or homogeneous Hilbert problems and various linear integral equation methods have been developed in detail and found different interesting applications. Recently many efforts of different authors were aimed at finding of generalizations of these solution generating methods to various (symmetry reduced) gravity, string gravity and supergravity models in four and higher dimensions. However, in some cases it occurred that even after the integrability of a system was evidenced, some difficulties arise which do not allow the authors to develop some effective methods for constructing of solutions. The present survey includes some remarks concerning the history of discoveries of some of the well known solution generating methods for these equations, brief descriptions of various approaches and their scopes as well as some comments concerning the possible difficulties of generalizations of various approaches to more complicate gravity models and possible ways for avoiding these difficulties., Comment: 22 pages, based on the opening remarks given at the MGAT1 parallel session of the Twelfth Marcel Grossmann Meeting (Paris, July 12 - 18, 2009)

The necessity of this Comment was invoked by numerous mistakes, erroneous discussions and misleading citations curiously collected in the paper of F.J.Ernst, V.S.Manko and E.Ruiz and concerning the interrelations between two integral equation methods developed for solution of Einstein - Maxwell equations more than twenty five years ago. At first, we clarify the origin of the errors in the paper of F.J.Ernst, V.S.Manko and E.Ruiz which gave rise to so curious authors "conclusions" as that the monodromy transform integral equations "...are simple combinations of Sibgatullin's integral equations and normalizing conditions..." or even that "...in the electrovac case Alekseev's integral equations are erroneous...". In the Comment, the way of correct derivation of Sibgatullin's reduction of the Hauser and Ernst integral equations in the context of the monodromy transform approach is briefly outlined. In response to various speculations and priority claims collected in the section 3 of the F.J.Ernst, V.S.Manko and E.Ruiz paper, the concrete references are given here to the papers which were ignored completely by these authors and which show that the so called "extended electrovacuum N-soliton solutions" considered by E.Ruiz, V.S. Manko and J. Martin in 1995, are not new because all these solutions are the particular cases of a larger class of solutions constructed much earlier in explicit (determinant) form using the monodromy transform equations, and that the real story of construction of the solution for superposition of fields of two Reissner - Nordstr\"om sources and of corresponding equilibrium configurations found in our papers with V.Belinski differs crucially from that, which one can read in the paper of F.J.Ernst, V.S.Manko and E.Ruiz., Comment: 11 pages, no figures

Integrable structure of the symmetry reduced dynamics of massless bosonic sector of the heterotic string effective action is presented. For string background equations that govern in the space-time of $D$ dimensions ($D\ge 4$) the dynamics of interacting gravitational, dilaton, antisymmetric tensor and any number $n\ge 0$ of Abelian vector gauge fields, all depending only on two coordinates, we construct an \emph{equivalent} $(2 d+n)\times(2 d+n)$ matrix spectral problem ($d=D-2$). This spectral problem provides the base for the development of various solution constructing procedures (dressing transformations, integral equation methods). For the case of the absence of Abelian gauge fields, we present the soliton generating transformations of any background with interacting gravitational, dilaton and the second rank antisymmetric tensor fields. This new soliton generating procedure is available for constructing of various types of field configurations including stationary axisymmetric fields, interacting plane, cylindrical or some other types of waves and cosmological solutions., Comment: 4 pages; added new section on Belinski-Zakharov solitons and new expressions for calculation of the conformal factor; corrected typos

In this paper we present a 5-parametric family of static asymptotically flat solutions for the superposed gravitational and electromagnetic fields of two Reissner-Nordstr\"om sources with arbitrary parameters -- masses, charges and separating distance. A procedure for solving of the linear singular integral equation form of the electrovacuum Einstein - Maxwell equations for stationary axisymmetric fields is described in detail. The 4-parametric family of equilibrium configurations of two Reissner-Nordstr\"om sources (one of which should be a black hole and another one -- a naked singularity) presented in our recent paper \cite{Alekseev-Belinski:2007} arises after a restriction of the parameters of the 5-parametric solution presented here by the equilibrium condition which provides the absence in the solution of conical points on the symmetry axis between the sources., Comment: 24 pages, submitted to the Proceedings of the Eleventh Marcel Grossmann Meeting (Berlin, July 23 - 29, 2006)

A brief sketch of the formulation of the monodromy transform approach and corresponding integral equation methods as well as of various applications of this approach for solution of integrable symmetry reductions of Einstein's field equations is presented., Comment: 3 pages, talk given at the Eleventh Marcel Grossmann Meeting (Berlin, July 23 - 29, 2006), submitted to MG11 Proceedings

An asymptotically flat static solution of Einstein-Maxwell equations which describes the field of two non-extreme Reissner - Nordstr\"om sources in equilibrium is presented. It is expressed in terms of physical parameters of the sources (their masses, charges and separating distance). Very simple analytical forms were found for the solution as well as for the equilibrium condition which guarantees the absence of any struts on the symmetry axis. This condition shows that the equilibrium is not possible for two black holes or for two naked singularities. However, in the case when one of the sources is a black hole and another one is a naked singularity, the equilibrium is possible at some distance separating the sources. It is interesting that for appropriately chosen parameters even a Schwarzschild black hole together with a naked singularity can be "suspended" freely in the superposition of their fields., Comment: 4 pages; accepted for publication in Phys. Rev. D

For the fields depending on two of the four space-time coordinates only, the spaces of local solutions of various integrable reductions of Einstein's field equations are shown to be the subspaces of the spaces of local solutions of the ``null-curvature'' equations constricted by a requirement of a universal (i.e. solution independent) structures of the canonical Jordan forms of the unknown matrix variables. These spaces of solutions of the ``null-curvature'' equations can be parametrized by a finite sets of free functional parameters -- arbitrary holomorphic (in some local domains) functions of the spectral parameter which can be interpreted as the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. Direct and inverse problems of such mapping (``monodromy transform''), i.e. the problem of finding of the monodromy data for any local solution of the ``null-curvature'' equations with given canonical forms, as well as the existence and uniqueness of such solution for arbitrarily chosen monodromy data are shown to be solvable unambiguously. The linear singular integral equations solving the inverse problems and the explicit forms of the monodromy data corresponding to the spaces of solutions of the symmetry reduced Einstein's field equations are derived., Comment: LaTeX, 33 pages, 1 figure. Typos, language and reference corrections

The integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric $d\times d$-matrix Ernst potentials are elucidated. These equations arise in the string theory as the equations of motion for a truncated bosonic parts of the low-energy effective action respectively for a dilaton and $d\times d$ - matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, U(1) gauge vector field and an antisymmetric 3-form field, all depending on two space-time coordinates only. We construct the corresponding spectral problems based on the overdetermined $2d\times 2d$-linear systems with a spectral parameter and the universal (i.e. solution independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed conditions of existence for each of these systems of two matrix integrals with appropriate symmetries provide a specific (coset) structures of the related matrix variables. An equivalence of these spectral problems to the original field equations is proved and some approach for construction of multiparametric families of their solutions is envisaged., Comment: 15 pages, no figures, LaTeX; based on the talk given at the Workshop ``Nonlinear Physics: Theory and Experiment. III'', 24 June - 3 July 2004, Gallipoli (Lecce), Italy. Minor typos, language and references corrections. To be published in the proceedings in Theor. Math. Phys

We consider the collisions of plane gravitational and electromagnetic waves with distinct wavefronts and of arbitrary polarizations in a Minkowski background. We first present a new, completely geometric formulation of the characteristic initial value problem for solutions in the wave interaction region for which initial data are those associated with the approaching waves. We present also a general approach to the solution of this problem which enables us in principle to construct solutions in terms of the specified initial data. This is achieved by re-formulating the nonlinear dynamical equations for waves in terms of an associated linear problem on the spectral plane. A system of linear integral ``evolution'' equations which solve this spectral problem for specified initial data is constructed. It is then demonstrated explicitly how various colliding plane wave space-times can be constructed from given characteristic initial data., Comment: 33 pages, 3 figures, LaTeX. Accepted for publication in Classical and Quantum Gravity

The generalized Einstein - Maxwell field equations which arise from a truncated bosonic part of the low - energy string gravity effective action in four dimensions (the so called Einstein-Maxwell - axion - dilaton theory) are considered. The integrable structure of these field equations for D=4 space-times with two commuting isometries is elucidated. We express the dynamical part of the reduced equations as integrability conditions of some overdetermined $4\times 4$-matrix linear system with a spectral parameter. The remaining part of the field equations are expressed as the conditions of existence for this linear system of two $4\times 4$-matrix integrals of special structures. This provides a convenient base for a generalization to these equations of various solution generating methods developed earlier in General Relativity., Comment: 8 pages, LaTeX. Based on the talk given at the Workshop ``Supersymmetry and Quantum Symmetries'' (JINR, Dubna, July 2003)

A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein's equations for space-times with two commuting isometries in General Relativity and in some string theory induced gravity models. In all cases the associated linear systems of similar structures are used, and their fundamental solutions admit an alternative representations by two ``scattering'' matrices of a simple analytical structures on the spectral plane. The condition of equivalence of these representations leads to the linear ``integral evolution equations'' whose scalar kernels and right hand sides are determined completely by the initial data for the fields specified on the two initial characteristics. If the initial data for the fields are given, all field components of the corresponding solution can be expressed in quadratures in terms of a unique solution of these quasi - Fredholm integral evolution equations., Comment: 8 pages, RevTeX 4; as submitted to the Proceedings of the workshop "Nonlinear Physics: Theory and Experiment. II" (Gallipili (Lecce), Italy, 27.06.2002 - 06.07.2002), World Scientific, Singapore

A new development of the ``monodromy transform'' method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (``scattering'') matrices give rise to a new set of linear (quasi-Fredholm) integral equations equivalent to the symmetry reduced Einstein equations. Unlike previously derived singular integral equations constructed with the use of conserved (nonevolving) monodromy data on the spectral plane for the fundamental solutions of associated linear systems, the scalar kernels of the new equations include another kind of functional parameters -- the evolving (``dynamical'') monodromy data for the scattering matrices. For hyperbolic reductions, in the context of characteristic initial value problem these data are determined completely by the characteristic initial data for the fields. In terms of solutions of the new integral equations the field components are expressed in quadratures., Comment: LaTeX, 23 pages; v2: the third displayed equation in sec. 4.6 was corrected and the reference [25] was adjusted

A method is presented for solving the characteristic initial value problem for the collision and subsequent nonlinear interaction of plane gravitational or gravitational and electromagnetic waves in a Minkowski background. This method generalizes the monodromy transform approach to fields with nonanalytic behaviour on the characteristics inherent to waves with distinct wave fronts. The crux of the method is in a reformulation of the main nonlinear symmetry reduced field equations as linear integral equations whose solutions are determined by generalized (``dynamical'') monodromy data which evolve from data specified on the initial characteristics (the wavefronts)., Comment: 4 pages, RevTeX

For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so called `monodromy transform' approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases., Comment: 7 pages, minor correction and reduction to conform with published version

In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called "monodromy transform approach", which provides some general base for the study of various integrable space - time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented., Comment: 8 pages, 1 figure, LaTeX2e; based on a talk given at the International Conference 'Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives', (Landau Institute for Theoretical Physics, Chernogolovka, Moscow region, Russia, August 3 -- 10, 1999); as submitted to Physica D

The approach, referred to as "monodromy transform", provides some general base for solution of all known integrable space - time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key points of this approach, applied to Einstein equations for vacuum and to Einstein - Maxwell equations for electrovacuum fields in the cases, reducible to the known Ernst equations. Definition of the monodromy data, formulation and solution of the direct and inverse problems of the monodromy transform, a proof of existence and uniqueness of their solutions, the structure of the basic linear singular integral equations and their regularizations, which lead to the equations of (quasi-)Fredholm type are also discussed. A construction of general local solution of these equations is given in terms of homogeneously convergent functional series., Comment: 4 pages, 1 figure, LaTeX2e; based on a talk given at the `International European Conference on Gravitation: Journ\'ees Relativistes 99.' Weimar, Germany, 12 - 17 Sep 1999; as submitted to Annalen der Physik (Leipzig)

A monodromy transform approach, presented in this communication, provides a general base for solution of space-time symmetry reductions of Einstein equations in all known integrable cases, which include vacuum, electrovacuum, massless Weyl spinor field and stiff matter fluid, as well as some string theory induced gravity models. It was found a special finite set of functional parameters, defined as the monodromy data for the fundamental solution of associated spectral problem. Similarly to the scattering data in the inverse scattering transform, these monodromy data can be used for characterization of any local solution of the field equations. A "direct" and "inverse" problems of such monodromy transform admit unambiguous solutions. For the linear singular integral equation with a scalar (i.e. non-matrix) kernel, which solves the inverse problem of this monodromy transform, an equivalent regularization -- a Fredholm linear integral equation of the second kind is constrcuted in several convenient forms. For arbitrary choice of the monodromy data a simple iterative method leads to an effective construction of the solution in terms of homogeneously convergent functional series., Comment: 7 pages, 1 figure, LaTeX2e; as submitted to the Proceedings of the Workshop "Nonlinearity Integrability and All That. Twenty Years After NEEDS'79", Gallipoli (Lecce), Italy - July 1 - July 10, 1999)

Exact solutions of Einstein's vacuum equations are considered which describe gravitational waves with distinct wavefronts. A family of such solutions presented recently in which the wavefronts have various geometries and which propagate into a number of physically significant backgrounds is here related to an integral representation which is a generalisation of the Rosen pulse solution for cylindrical waves. A nondiagonal solution is also constructed which is a generalisation of the Rosen pulse, being a cylindrical pulse wave with two states of polarization propagating into a Minkowski background. The solution is given in a complete and explicit form. A further generalisation to include electromagnetic waves with a distinct wavefront of the same type is also discussed., Comment: 14 pages, Plain TeX, no figures. To appear in Classical and Quantum Gravity