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

TL;DR

This paper explores the geometric structure of second-order PDEs in 3D, revealing how Einstein-Weyl conditions can be used to test integrability and classify such equations.

Contribution

It demonstrates that the covector w in Einstein-Weyl geometry can be expressed in terms of the PDE, providing a new integrability test and explicit dispersionless Lax pairs.

Findings

w is expressible in terms of the PDE for generic equations

Provides an explicit formula for dispersionless Lax pairs

Partial classification results for PDEs with Einstein-Weyl 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.