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

@article{Dimakis2011TheNC,
  title={The non-autonomous chiral model and the Ernst equation of General Relativity in the bidifferential calculus framework},
  author={Aristophanes Dimakis and Nils Kanning and Folkert Muller-Hoissen},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2011},
  volume={7},
  pages={118}
}
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… 
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7 Citations

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62 References

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