The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

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 Ein...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2011
Main Authors: Dimakis, A., Kanning, N., Müller-Hoissen, F.
Format: Article
Language:English
Published: Інститут математики НАН України 2011
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/148083
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework / A. Dimakis, N. Kanning, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 57 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-148083
record_format dspace
spelling Dimakis, A.
Kanning, N.
Müller-Hoissen, F.
2019-02-16T20:47:24Z
2019-02-16T20:47:24Z
2011
The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework / A. Dimakis, N. Kanning, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 57 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 37K10; 16E45
DOI: http://dx.doi.org/10.3842/SIGMA.2011.118
https://nasplib.isofts.kiev.ua/handle/123456789/148083
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.
We would like to thank Vladimir S. Manko and anonymous referees for helpful comments. During the course of this work, N.K. has been at the Max-Planck-Institute for Dynamics and Self-Organization in G¨ottingen.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
spellingShingle The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
Dimakis, A.
Kanning, N.
Müller-Hoissen, F.
title_short The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
title_full The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
title_fullStr The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
title_full_unstemmed The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
title_sort non-autonomous chiral model and the ernst equation of general relativity in the bidifferential calculus framework
author Dimakis, A.
Kanning, N.
Müller-Hoissen, F.
author_facet Dimakis, A.
Kanning, N.
Müller-Hoissen, F.
publishDate 2011
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description 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.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/148083
citation_txt The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework / A. Dimakis, N. Kanning, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 57 назв. — англ.
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first_indexed 2025-11-29T03:43:47Z
last_indexed 2025-11-29T03:43:47Z
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