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

Author:

Berjawi S.1,Ferapontov E. V.12ORCID,Kruglikov B.3,Novikov V.1

Affiliation:

1. Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK

2. Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa, Russia

3. Department of Mathematics and Statistics, UiT the Arctic University of Norway, Tromso 9037, Norway

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.

Funder

Engineering and Physical Sciences Research Council

Trond Mohn Foundation

Troms\o\ Research Foundation

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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1. Deformations of dispersionless Lax systems *;Classical and Quantum Gravity;2023-11-07

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3. Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation;Differential Geometry and its Applications;2021-06

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