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Second-order PDEs in four dimensions with half-flat conformal structure
journal contribution
posted on 2020-01-06, 15:36 authored by Sobhi Berjawi, Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov, Vladimir NovikovVladimir NovikovWe study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.
Funding
EPSRC grant EP/N031369/1
Trond Mohn Foundation
Tromsø Research Foundation
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School
- Science
Department
- Mathematical Sciences
Published in
Proceedings of the Royal Society A: Mathematical, Physical and Engineering SciencesVolume
476Issue
2233Publisher
The Royal SocietyVersion
- AM (Accepted Manuscript)
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© The AuthorsPublisher statement
This paper : Berjawi S, Ferapontov EV, Kruglikov B, Novikov V. 2020 Second-order PDEs in four dimensions with half-flat conformal structure. Proc. R. Soc. A 476: 20190642 was accepted for publication in the journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences and the definitive published version is available at https://doi.org/10.1098/rspa.2019.0642.Acceptance date
2019-12-18Publication date
2020-01-29Copyright date
2020ISSN
1364-5021eISSN
1471-2946Publisher version
Language
- en
Depositor
Prof Evgeny Ferapontov. Deposit date: 29 December 2019Article number
20190642Usage metrics
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