Adler–Oevel-Ragnisco type operators and Poisson vertex algebras

The theory of triples of Poisson brackets and related integrable systems, based on a classical $R$-matrix $R \in \mathrm{End}_\mathbb{F}(\mathfrak{g})$, where $\mathfrak{g}$ is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel–Ragnisco and Li–Parmentier [$\href{https://doi.org/10.1016/0378-4371(89)90398-1}{\textrm{OR89}}$, $\href{https://doi.org/10.1007/BF01228340}{\textrm{LP89}}$]. 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.