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Gravity World. GAA. Gallipoli 2008. GAA. Nonlinear phenomena in strong gravitational fields. with two-dimensional symmetries. Vacuum. Symmetries:. --- Non-vacuum integrable reductions of Einstein’s field equations. Thirty years of solitons in General Relativity:. Integrability - ?.

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## Gravity World

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**Gravity**World GAA**Gallipoli 2008**GAA**Nonlinear phenomena in strong gravitational fields**with two-dimensional symmetries**Vacuum**Symmetries: --- Non-vacuum integrable reductions of Einstein’s field equations Thirty years of solitons in General Relativity: Integrability - ? -- R.Geroch –conjecture (1972) -- W.Kinnersley –inf.dim. algebra (1977…) -- D.Maison - Lax pair +conjecture (1978) Belinski and Zakharov (1978)-- Inverse Scattering Method -- Soliton solutions on arbit. backgr. -- Riemann – Hilbert problem + linear singular integral equations Later results: -- Backlund transformations (Harrison 1978, Neugebauer 1979) -- Homogeneous Hilbert problem (Hauser & Ernst, 1979+Sibgatullin 1984) -- Monodromy Transform + linera singular integral equations (GA 1985) -- Finite-gap solutions (Korotkin&Matveev 1987, Neugebauer&Meinel 1993) -- Charateristic initial value problem (GA & Griffiths 2001)**Complete integrability and monodromy preserving structure**of the symmetry reduced heterotic string effective equations in any dimensions Gallipoli 2008 G. Alekseev Bosonic sector of heterotic string effective action: The symmetry ansatz:**Bosonic action in the Einstein frame:**Field equations:**Dynamical degrees of freedom:**Conformal factor: Geometrically defined coordinates and :**Matrix Ernst-like dynamical variables:**Matrix Ernst-like form of the dynamical equations Examples for the choice of**Associated linear system for N x N matrices**Coordinates:**1)**Monodromy transform approach Analytical structure of on w – plane: GA, Sov. Phys (1985) ; 1)**Monodromy data for solutions:**Monodromy data of a given solution**1)**Inverse problem of the monodromy transform 1) GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005**Solutions for analitically matched, rational monodromy data**Generic data: Analytically matched data: Unknowns: Rational, analytically matched data:**Equilibrium configurations of two charged masses**In equilibrium: 1) GA and V.Belinski Phys.Rev. D (2007)**1)**Inverse problem of the monodromy transform Free space of the monodromy data Space of solutions Theorem 1. For any holomorphic local solution near , Is holomorphic on and the ``jumps’’ of on the cuts satisfy the H lder condition and are integrable near the endpoints. posess the same properties 1) GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys. 2005***)**Theorem 2. For any holomorphic local solution near , possess the local structures and where are holomorphic on respectively. Fragments of these structures satisfy in the algebraic constraints (for simplicity we put here ) and the relations in boxes give rise later to the linear singular integral equations. *) In the case N-2d we do not consider the spinor field and put**Theorem 3.***) For any local solution of the ``null curvature'' equations with the above Jordan conditions, the fragments of the local structures of and on the cuts should satisfy where the dot for N=2d means a matrix product and the scalar kernels (N=2,3) or dxd-matrix (N=2d) kernels and coefficients are where and each of the parameters and runs over the contour ; e.g.: In the case N-2d we do not consider the spinor field and put *)**Theorem 4.**For arbitrarily chosen extended monodromy data – the scalar functions and two pairs of vector (N=2,3) or only two pairs of dx2d and 2dxd – matrix (N=2d) functions and holomorphic respectively in some neighbor-- hoods and of the points and on the spectral plane, there exists some neighborhood of the initial point such that the solutions and of the integral equations given in Theorem 3 exist and are unique in and respectively. The matrix functions and are defined as is a normalized fundamental solution of the associated linear system with the Jordan conditions.

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