Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property

Abstract

We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampere equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations.

In this version: (1) We correct an error in the proof of Lemma from Sec.2.3 due to an index shift; (2) We extend the characterisation of Monge-Ampere property for PDE in implicit form, in addition to evolutionary form; (3) We prove that integrability in higher dimensions also implies the Monge-Ampere property. The ancillary files should be accessed via the previous version, arXiv:1707.08070v1