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Calculations needed for the proof of Theorem 1 (Mathematica file Computation Th1.nb and its description Computation Th1.pdf) from Second-order PDEs in four dimensions with half-flat conformal structure

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posted on 2020-01-02, 08:40 authored by S. Berjawi, E.V. Ferapontov, B. Kruglikov, V. Novikov
We study second-order 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 nontrivial 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.

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