Dispersionless integrable systems in 3D and Einstein-Weyl geometry

TL;DR

This paper establishes a geometric criterion linking the integrability of certain dispersionless PDEs in 3D to their associated Einstein-Weyl structures, revealing a deep connection between PDE integrability and differential geometry.

Contribution

It demonstrates that the integrability of second order dispersionless PDEs is characterized by their formal linearizations defining Einstein-Weyl structures in 3D.

Findings

Symbols of formal linearizations define Einstein-Weyl structures in 3D

Integrability corresponds to the Einstein-Weyl condition in the geometric structure

Geometric perspective provides a new criterion for PDE integrability

Abstract

For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of these dispersionless PDEs can be seen from the geometry of their formal linearizations.