Characteristic integrals in 3D and linear degeneracy

TL;DR

This paper explores characteristic integrals in three-dimensional PDEs, revealing their parametrization by points on the Veronese variety for certain linearly degenerate integrable equations, thus extending 2D theory.

Contribution

It introduces the concept of characteristic integrals in 3D and links them to the Veronese variety for a class of integrable PDEs, advancing the understanding of their structure.

Findings

Characteristic integrals in 3D are parametrized by the Veronese variety.

The study extends the theory of characteristic conservation laws from 2D to 3D.

Identifies a geometric structure underlying integrable PDEs in three dimensions.

Abstract

Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that, for a class of second-order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parametrised by points on the Veronese variety.