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17 September 2019
 
  » arxiv » gr-qc/0503043

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Monodromy data parametrization of the spaces of local solutions of integrable reductions of Einstein's field equations
G. A. Alekseev ;
Date 9 Mar 2005
Journal Theor.Math.Phys. 143 (2005) 720-740; Teor.Mat.Fiz. 143 (2005) 278-304
Subject General Relativity and Quantum Cosmology; Exactly Solvable and Integrable Systems | gr-qc hep-th nlin.SI
AbstractFor 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. The spaces of local solutions of these constraint ``null curvature’’ equations can be parametrized by a finite set of holomorphic functions of the spectral parameter which can be interpreted as a complete set of the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. The 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 problem are derived. The explicit forms of the monodromy data corresponding to the spaces of solutions of Einstein’s field equations are determined.
Source arXiv, gr-qc/0503043
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