Second-order PDEs in 4D with half-flat conformal structure
Differential Geometry
2020-02-04 v1 Mathematical Physics
math.MP
Exactly Solvable and Integrable Systems
Abstract
We study second-order PDEs in 4D 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-Ampere 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-Ampere type. Some partial classification results of Monge-Ampere equations in 4D with half-flat conformal structure are also obtained.
Cite
@article{arxiv.2002.00373,
title = {Second-order PDEs in 4D with half-flat conformal structure},
author = {Sobhi Berjawi and Eugene Ferapontov and Boris Kruglikov and Vladimir Novikov},
journal= {arXiv preprint arXiv:2002.00373},
year = {2020}
}