Adler-Oevel-Ragnisco type operators and Poisson vertex algebras

TL;DR

This paper extends the theory of Poisson brackets and integrable systems to an affine setting by introducing continuous Poisson vertex algebras and Adler-type identities, advancing the understanding of Hamiltonian PDE hierarchies.

Contribution

It develops an affine analogue of classical Poisson bracket triples using continuous Poisson vertex algebras and constructs Adler-type identities for integrability analysis.

Findings

Introduced continuous Poisson vertex algebras.

Constructed triples of Poisson lambda-brackets.

Applied Adler identities to Hamiltonian PDE hierarchies.

Abstract

The theory of triples of Poisson brackets and related integrable systems, based on a classical R-matrix R in End_F(g), where g is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel-Ragnisco and Li-Parmentier [OR89,LP89]. In the present paper we develop an "affine" analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson lambda-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.