Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Abstract

Einstein–Weyl geometry is a triple ({mathbb {D}},g,omega ) where {mathbb {D}} is a symmetric connection, [g] is a conformal structure and omega is a covector such that bullet connection {mathbb {D}} preserves the conformal class [g], that is, {mathbb {D}}g=omega g; bullet trace-free part of the symmetrised Ricci tensor of {mathbb {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 omega 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 omega is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and omega 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.