Veronese webs and nonlinear PDEs

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.

Cited by in corpus (8)

• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Infinitely many nonlocal conservation laws for the $ABC$ equation with $A+B+C\ne 0$
• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• On deformations of the dispersionless Hirota equation
• Dispersionless integrable hierarchies and GL(2,R) geometry
• GL(2)-geometry and complex structures
• Second-order PDEs in 4D with half-flat conformal structure
• Lagrangian multiforms and dispersionless integrable systems