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 TMF, 1988, Volume 77, Number 1, Pages 25–41 (Mi tmf5678)

Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum

D. A. Korotkin

Abstract: A new and large class of exact solutions of the stationary axisymmetric Einstein equation, which are expressed in terms of the Riemann $\theta$ function, is constructed. The properties of the constructed “finite-gap” solutions differ significantly from those of the well-known finite-gap solutions (for example, of the Korteweg–de Vries equation and the nonlinear Schrödinger equation). In particular, the dependence on the dynamical variables in the final expressions is given by a trajectory on a manifold of moduli of algebraic curves, and not on the Jacobi manifold of a given curve. In a degenerate case the constructed solutions include all the main known solutions that can be expressed in terms of elementary functions.

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English version:
Theoretical and Mathematical Physics, 1988, 77:1, 1018–1031

Bibliographic databases:

Citation: D. A. Korotkin, “Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum”, TMF, 77:1 (1988), 25–41; Theoret. and Math. Phys., 77:1 (1988), 1018–1031

Citation in format AMSBIB
\Bibitem{Kor88} \by D.~A.~Korotkin \paper Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum \jour TMF \yr 1988 \vol 77 \issue 1 \pages 25--41 \mathnet{http://mi.mathnet.ru/tmf5678} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=972479} \transl \jour Theoret. and Math. Phys. \yr 1988 \vol 77 \issue 1 \pages 1018--1031 \crossref{https://doi.org/10.1007/BF01028676} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1988AD89600003} 

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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. D. A. Korotkin, “Finite-gap solutions of self-duality equations for $SU(1,1)$ and $SU(2)$ groups and their axisymmetric stationary reductions”, Math. USSR-Sb., 70:2 (1991), 355–366
2. D. A. Korotkin, V. B. Matveev, “Theta Function Solutions of the Schlesinger System and the Ernst Equation”, Funct. Anal. Appl., 34:4 (2000), 252–264
3. C. Klein, “Exact Relativistic Treatment of Stationary Counter-Rotating Dust Disks: Axis, Disk, and Limiting Cases”, Theoret. and Math. Phys., 127:3 (2001), 767–778
4. C. Klein, “Isomonodromy Approach to Boundary Value Problems for the Ernst Equation”, Theoret. and Math. Phys., 134:1 (2003), 72–85
5. C. Klein, “The Kerr Solution on Partially Degenerate Hyperelliptic Riemann Surfaces”, Theoret. and Math. Phys., 137:2 (2003), 1520–1526
6. Klein, C, “On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks”, Annalen der Physik, 12:10 (2003), 599
7. Matveev, VB, “30 years of finite-gap integration theory”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 837
8. Aristophanes Dimakis, Nils Kanning, Folkert Müller-Hoissen, “The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework”, SIGMA, 7 (2011), 118, 27 pp.
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