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The non-autonomous chiral model and the Ernst equation of General Relativity in the bidifferential calculus framework

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Kanning,  Nils
Max Planck Institute for Dynamics and Self-Organization, Max Planck Society;

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

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Citation

Dimakis, A., Kanning, N., & Müller-Hoissen, F. (2011). The non-autonomous chiral model and the Ernst equation of General Relativity in the bidifferential calculus framework. Symmetry, Integrability and Geometry: Methods and Applications, 7: 118. doi:10.3842/SIGMA.2011.118.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0029-113B-1
Abstract
The non-autonomous chiral model equation for an m×m matrix function on a two-dimensional space appears in particular in general relativity, where for m=2 a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for m=3 solutions of the Einstein-Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Deminski-Newman metrics.