Integrability of dispersionless Hirota-type equations and the symplectic Monge-Ampère property
journal contributionposted on 2020-02-10, 11:57 authored by Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov, Vladimir NovikovVladimir Novikov
We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampère property in any dimension ≥ 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-Ampère 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.
EPSRC grant EP/N031369/1
- Mathematical Sciences