Berjawi2022_Article_Second-OrderPDEsIn3DWithEinste.pdf (552.22 kB)
Second-order PDEs in 3D with Einstein-Weyl conformal structure
journal contributionposted 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.
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 CouncilFind out more...
- Mathematical Sciences
Published inAnnales Henri Poincaré
Pages2579 - 2609
PublisherSpringer (part of Springer Nature)
- VoR (Version of Record)
Rights holder© The Authors
Publisher statementThis 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/