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New integral equation form of integrable reductions of Einstein equations  G.A.Alekseev
;  Date: 
28 May 2001  Journal:  Theor.Math.Phys. 129 (2001) 14661483; Teor.Mat.Fiz. 129 (2001) 184206  Subject:  General Relativity and Quantum Cosmology; Mathematical Physics; Exactly Solvable and Integrable Systems  grqc hepth mathph math.MP nlin.SI  Abstract:  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 (quasiFredholm) 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.  Source:  arXiv, grqc/0105111  Services:  Forum  Review  PDF  Favorites 


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