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Integrability of dispersionless Hirota-type equations and the symplectic Monge-Ampère property

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journal contribution
posted on 10.02.2020, 11:57 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.

Funding

EPSRC grant EP/N031369/1

History

School

  • Science

Department

  • Mathematical Sciences

Published in

International Mathematics Research Notices

Volume

2021

Issue

18

Pages

14220-14251

Publisher

Oxford University Press

Version

AM (Accepted Manuscript)

Rights holder

© The Authors

Publisher statement

This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record E V Ferapontov, B Kruglikov, V Novikov, Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property, International Mathematics Research Notices, 2021 (18), pp.14220-14251 is available online at: https://doi.org/10.1093/imrn/rnaa025.

Acceptance date

29/01/2020

Publication date

2020-02-29

Copyright date

2020

ISSN

1073-7928

eISSN

1687-0247

Language

en

Depositor

Dr Vladimir Novikov. Deposit date: 8 February 2020