Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 009, 31 pages      arXiv:1207.1308      https://doi.org/10.3842/SIGMA.2013.009

Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

Aristophanes Dimakis a and Folkert Müller-Hoissen b
a) Department of Financial and Management Engineering, University of the Aegean, 82100 Chios, Greece
b) Max-Planck-Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

Received November 12, 2012, in final form January 29, 2013; Published online February 02, 2013

Abstract
We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D−2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings).

Key words: bidifferential calculus; binary Darboux transformation; chiral model; Einstein equations; black ring.

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