Homogeneous Hamiltonian operators and the theory of coverings

TL;DR

This paper introduces a novel method based on differential coverings to connect PDE systems with Hamiltonian operators, revealing geometric conditions that deepen understanding of symmetries and conservation laws.

Contribution

It develops a new approach linking PDEs and Hamiltonian operators via differential coverings, with geometric insights into the structure of such systems.

Findings

Conditions on Hamiltonian operators derived from the method

Geometric interpretations of the operator-system relations

Application to quasilinear first-order PDEs

Abstract

A new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps symmetries/conserved quantities into symmetries/conserved quantities of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin-Novikov homogeneous Hamiltonian operator the method yields conditions on the operator and the system that have interesting differential and projective geometric interpretations.