Integrability via geometry: dispersionless differential equations in three and four dimensions

TL;DR

This paper establishes a geometric characterization of integrability for certain second order PDEs in three and four dimensions, linking dispersionless Lax pairs to Einstein-Weyl and self-dual conformal structures.

Contribution

It proves the equivalence between dispersionless Lax pairs with spectral parameter and specific geometric structures for nondegenerate hyperbolic PDEs in 3D and 4D.

Findings

Dispersionless Lax pairs correspond to Einstein-Weyl structures in 3D.

In 4D, they correspond to self-dual conformal structures.

The results apply to any PDE system with a quadric characteristic variety.

Abstract

We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.