Loughborough University
Browse
Berjawi2022_Article_Second-OrderPDEsIn3DWithEinste.pdf (552.22 kB)

Second-order PDEs in 3D with Einstein-Weyl conformal structure

Download (552.22 kB)
journal contribution
posted on 2021-12-07, 12:00 authored by Sobhi Berjawi, Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov, Vladimir NovikovVladimir Novikov
Einstein–Weyl geometry is a triple (D,g,ω) where D is a symmetric connection, [g] is a conformal structure and ω is a covector such that ∙ connection D preserves the conformal class [g], that is, Dg=ωg; ∙ trace-free part of the symmetrised Ricci tensor of D vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector ω is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector ω is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and ω provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

Funding

Russian Science Foundation No. 21-11-00006, https://rscf.ru/project/21-11-00006/

Trond Mohn Foundation and Tromsø Research Foundation

A novel approach to integrability of semi-discrete systems

Engineering and Physical Sciences Research Council

Find out more...

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Annales Henri Poincaré

Volume

23

Issue

7

Pages

2579 - 2609

Publisher

Springer (part of Springer Nature)

Version

  • VoR (Version of Record)

Rights holder

© The Authors

Publisher statement

This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Acceptance date

2021-11-18

Publication date

2021-12-07

Copyright date

2021

ISSN

1424-0637

eISSN

1424-0661

Language

  • en

Depositor

Prof Evgeny Ferapontov. Deposit date: 19 November 2021

Usage metrics

    Loughborough Publications

    Categories

    No categories selected

    Licence

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC