Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure
Peer reviewed, Journal article
Published version
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https://hdl.handle.net/11250/3058065Utgivelsesdato
2021Metadata
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Originalversjon
Berjawi, S., Ferapontov, E. V., Kruglikov, B. S., & Novikov, V. S. (2022, July). Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure. In Annales Henri Poincaré (Vol. 23, No. 7, pp. 2579-2609). Cham: Springer International Publishing. 10.1007/s00023-021-01140-2Sammendrag
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.