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TMF, 2005, Volume 143, Number 2, Pages 278–304 (Mi tmf1815)  

This article is cited in 8 scientific papers (total in 8 papers)

Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations

G. A. Alekseev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We show that for the fields depending on only two of the four space-time coordinates, the spaces of local solutions of various integrable reductions of Einstein's field equations are the subspaces of the spaces of local solutions of the “null-curvature” equations selected by universal (i.e., solution-independent conditions imposed on the canonical (Jordan) forms of the desired matrix variables. Each of these spaces of solutions can be parameterized 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. We show that both the direct and inverse problems of such a map, i.e., the problem of finding the monodromy data for any local solution of the null-curvature equations for the given Jordan forms and also of proving the existence and uniqueness of such a solution for arbitrary monodromy data, can be solved unambiguously (the “monodromy transform”). We derive the linear singular integral equations solving the inverse problem and determine the explicit forms of the monodromy data corresponding to the spaces of solutions of Einstein's field equations.

Keywords: Einstein's equations, string gravity, integrability, singular integral equations, monodromy

DOI: https://doi.org/10.4213/tmf1815

Full text: PDF file (370 kB)
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English version:
Theoretical and Mathematical Physics, 2005, 143:2, 720–740

Bibliographic databases:

Received: 09.09.2004

Citation: G. A. Alekseev, “Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations”, TMF, 143:2 (2005), 278–304; Theoret. and Math. Phys., 143:2 (2005), 720–740

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Filippov, AT, “Integrable models of (1+1)-dimensional dilaton gravity coupled to scalar matter”, Theoretical and Mathematical Physics, 146:1 (2006), 95  mathnet  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. V. De Alfaro, A. T. Filippov, “Dimensional reduction of gravity and relation between static states, cosmologies, and waves”, Theoret. and Math. Phys., 153:3 (2007), 1709–1731  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Gao, YJ, “Inverse scattering method and soliton double solution family for the general symplectic gravity model”, Journal of Mathematical Physics, 49:8 (2008), 083506  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Alekseev G., Belinski V., “Einstein-Maxwell Solitons”, Cosmology and Gravitation, AIP Conference Proceedings, 1132, 2009, 333–385  crossref  adsnasa  isi  scopus  scopus
    5. V. de Alfaro, A. T. Filippov, “Multiexponential models of $(1+1)$-dimensional dilaton gravity and Toda–Liouville integrable models”, Theoret. and Math. Phys., 162:1 (2010), 34–56  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. A. T. Filippov, “Weyl–Eddington–Einstein affine gravity in the context of modern cosmology”, Theoret. and Math. Phys., 163:3 (2010), 753–767  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    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  crossref  adsnasa  isi  elib  scopus  scopus
    8. Fuchs A., Reisenberger M.P., “Integrable Structures and the Quantization of Free Null Initial Data For Gravity”, Class. Quantum Gravity, 34:18 (2017), 185003  crossref  mathscinet  zmath  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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