On a class of third-order nonlocal Hamiltonian operators

TL;DR

This paper classifies a specific class of third-order nonlocal Hamiltonian operators using Poisson vertex algebra theory, focusing on their geometric constraints and providing complete classifications for low-component cases.

Contribution

It introduces a classification of third-order nonlocal Hamiltonian operators based on differential-geometric conditions, expanding understanding of their structure.

Findings

Complete classification for 2-component operators

Complete classification for 3-component operators

Identification of geometric constraints for Hamiltonian operators

Abstract

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential-geometric constraints. Complete classification results in the 2-component and 3-component cases are obtained.