Binary Darboux transformations in bidifferential calculus and integrable 
reductions of vacuum Einstein equations

Dimakis, Aristophanes; Müller-Hoissen, Folkert

doi:10.3842/SIGMA.2013.009

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Binary Darboux transformations in bidifferential calculus and integrable reductions of vacuum Einstein equations

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Müller-Hoissen, Folkert
Max Planck Institute for Dynamics and Self-Organization, Max Planck Society;

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Dimakis, A., & Müller-Hoissen, F. (2013). Binary Darboux transformations in bidifferential calculus and integrable reductions of vacuum Einstein equations. Symmetry, Integrability and Geometry: Methods and Applications, 9: 009. doi:10.3842/SIGMA.2013.009.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0029-102D-9

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).