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

Abstract

Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the symmetrised Ricci tensor of D vanishes. Three-dimensional Einstein-Weyl structures arise naturally 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, the covector w 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-Ampere type), the covector w is also expressible in terms of the equation, thus providing an efficient dispersionless integrability test. The knowledge of g and w 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.

References in corpus (14)

• On integrability of (2+1)-dimensional quasilinear systems
• Einstein-Weyl geometry, the dKP equation and twistor theory
• A class of Einstein--Weyl spaces associated to an integrable system of hydrodynamic type
• Interpolating Dispersionless Integrable System
• On the Einstein-Weyl and conformal self-duality equations
• Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II
• Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs
• Integrable Background Geometries
• Veronese webs and nonlinear PDEs
• On a class of 2D integrable lattice equations
• Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy
• On deformations of the dispersionless Hirota equation
• Completely exceptional $2^{\text{nd}}$ order PDEs via conformal geometry and BGG resolution
• Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Cited by in corpus (3)

• Lagrangian multiforms and dispersionless integrable systems
• Recurrent Lorentzian Weyl spaces
• Hydrodynamic integrability via geometry