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

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.

SUBMITTER: Berjawi S

PROVIDER: S-EPMC7016558 | biostudies-literature | 2020 Jan

REPOSITORIES: biostudies-literature

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

Berjawi S S Ferapontov E V EV Kruglikov B B Novikov V V

Proceedings. Mathematical, physical, and engineering sciences 20200129 2233

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 dis ...[more]

PMID: 32082065

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