Veronese webs and nonlinear PDEs

TL;DR

This paper explores the deep connections between Veronese webs, integrable PDEs, and Einstein-Weyl structures, revealing new relations, deformations, and symmetries that enhance understanding of these geometric and integrable systems.

Contribution

It establishes links between Veronese webs and various integrable equations, introduces deformations preserving integrability, and connects finite-dimensional bi-Hamiltonian systems with dispersionless PDEs.

Findings

Relations between Veronese webs and integrable PDEs

Deformation of equations while preserving integrability

Construction of Einstein-Weyl structures from PDE solutions

Abstract

Veronese webs are closely related to bi-Hamiltonian systems, as was shown by Gelfand and Zakharevich. Recently a correspondence between Veronese three-dimensional webs and three-dimensional Einstein-Weyl structures of hyper-CR type was established. The latter were parametrized by Dunajski and Krynski via the solutions of the dispersionless Hirota equation. In this paper we show relations of Veronese three-dimensional webs to several other integrable equations, deform these equations preserving integrability via a dispersionless Lax pair and compute the corresponding contact symmetries, Backlund transformations and Einstein-Weyl structures. Realization of Veronese webs through solutions of these deformed integrable PDE is based on a correspondence between partially integrable Nijenhuis operators to the operator fields with vanishing Nijenhuis tensor. This correspondence could be used to construct a link between bi-Hamiltonian finite-dimensional integrable systems and dispersionless integrable PDE related to the Veronese webs.