Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs

TL;DR

This paper explores the geometric and analytical structures of Veronese webs, linking nonlinear wave equations, twistor transforms, and Riemann problems in three-dimensional cases, revealing deep connections between geometry and integrable PDEs.

Contribution

It establishes a novel connection between Veronese webs, nonlinear wave equations, and the nonlinear Riemann problem, providing new insights into their geometric and analytical properties.

Findings

Veronese webs in 3D relate to a special nonlinear wave equation.

Symmetries of Veronese webs correspond to Bäcklund--Darboux transformations.

Solutions to the wave equations can be characterized via the nonlinear Riemann problem.

Abstract

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web reveal themselves as B\"acklund--Darboux transformations between these non-linear wave equations. On the other hand, the twistor transform identifies Veronese webs with moduli spaces of rational curves on certain complex surfaces. These moduli spaces can be described in terms of the non-linear Riemann problem. This reduces solutions of these non-linear wave equations to the non-linear Riemann problem. We examine these relationships in the particular case of 3-dimensional Veronese webs, simultaneously investigating how these notions relate to general notions of geometry of webs.