Non-local Poisson structures and applications to the theory of integrable systems

TL;DR

This paper develops a rigorous framework for non-local Poisson structures using non-local Poisson vertex algebras and applies it to establish integrability of various evolution and hyperbolic equations via the Lenard-Magri scheme.

Contribution

It introduces a formal theory of non-local Poisson structures and demonstrates their application to integrability of complex differential equations.

Findings

Established conditions for applying the Lenard-Magri scheme to non-local Poisson structures.

Proved integrability of several evolution equations using the developed framework.

Demonstrated the applicability to hyperbolic equations.

Abstract

We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard-Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations.