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 TMF, 2001, Volume 129, Number 2, Pages 184–206 (Mi tmf529)

A New Integral Equation Form of Integrable Reductions of the Einstein Equations

G. A. Alekseev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We further develop the monodromy transformation method for analyzing hyperbolic and elliptic integrable reductions of the Einstein equations. The compatibility conditions for alternative representations of solutions of the associated linear systems with a spectral parameter in terms of a pair of dressing (“scattering”) matrices yield a new set of linear (quasi-Fredholm) integral equations that are equivalent to the symmetry-reduced Einstein equations. In contrast to the previously derived singular integral equations constructed using conserved (nonevolving) monodromy data for fundamental solutions of the associated linear systems, the scalar kernels of the new equations involve functional parameters of a different type, the evolving (“dynamic”) monodromy data for scattering matrices. In the context of the Goursat problem, these data are completely determined for hyperbolic reductions by the characteristic initial data for the fields. The field components are expressed in quadratures in terms of solutions of the new integral equations.

DOI: https://doi.org/10.4213/tmf529

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English version:
Theoretical and Mathematical Physics, 2001, 129:2, 1466–1483

Bibliographic databases:

Citation: G. A. Alekseev, “A New Integral Equation Form of Integrable Reductions of the Einstein Equations”, TMF, 129:2 (2001), 184–206; Theoret. and Math. Phys., 129:2 (2001), 1466–1483

Citation in format AMSBIB
\Bibitem{Ale01} \by G.~A.~Alekseev \paper A New Integral Equation Form of Integrable Reductions of the Einstein Equations \jour TMF \yr 2001 \vol 129 \issue 2 \pages 184--206 \mathnet{http://mi.mathnet.ru/tmf529} \crossref{https://doi.org/10.4213/tmf529} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1904793} \zmath{https://zbmath.org/?q=an:1034.83004} \transl \jour Theoret. and Math. Phys. \yr 2001 \vol 129 \issue 2 \pages 1466--1483 \crossref{https://doi.org/10.1023/A:1012822904758} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000173055900002} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Alekseev, GA, “Solving the characteristic initial-value problem for colliding plane gravitational and electromagnetic waves”
2. Alekseev G.A., Griffiths J.B., “Solving the characteristic initial-value problem for colliding plane gravitational and electromagnetic waves”
3. Alekseev, GA, “Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem”, Classical and Quantum Gravity, 21:23 (2004), 5623
4. Kechkin, OV, “Sigma-models coupled to gravity in string theory”, Physics of Particles and Nuclei, 35:3 (2004), 383
5. Karas, V, “Gravitating discs around black holes”, Classical and Quantum Gravity, 21:7 (2004), R1
6. Tongas, A, “Generalized hyperbolic Ernst equations for an Einstein-Maxwell-Weyl field”, Journal of Physics A-Mathematical and General, 38:4 (2005), 895
7. Alekseev G.A., “Monodromy Transform and the Integral Equation Method for Solving the String Gravity and Supergravity Equations in Four and Higher Dimensions”, Phys. Rev. D, 88:2 (2013), 021503
8. Alekseev G., “Travelling Waves in Expanding Spatially Homogeneous Space-Times”, Class. Quantum Gravity, 32:7 (2015), 075009
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