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Second-order PDEs in four dimensions with half-flat conformal structure

Permanent link
https://hdl.handle.net/10037/30904
DOI
https://doi.org/10.1098/rspa.2019.0642
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Date
2020-01-29
Type
Journal article
Tidsskriftartikkel
Peer reviewed

Author
Berjawi, S.; Ferapontov, Eugene V.; Kruglikov, Boris; Novikov, Vladimir S
Abstract
We 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.
Publisher
The Royal Society
Citation
Berjawi, Ferapontov, Kruglikov, Novikov. Second-order PDEs in four dimensions with half-flat conformal structure. Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. 2020;476(2233)
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