On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

Abstract

The theory of Poisson vertex algebras (PVAs) (Barakat et al. in Jpn J Math 4(2):141–252, 2009) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair $${(\mathcal{A},\{\cdot_\lambda\cdot\})}$$ of a differential algebra $${\mathcal{A}}$$ and a bilinear operation called the $${\lambda}$$ -bracket. We extend the definition to the class of algebras $${\mathcal{A}}$$ endowed with $${d \geq 1}$$ commuting derivations. We call this structure a multidimensional PVA: it is a suitable setting to study Hamiltonian PDEs with d spatial dimensions. We apply this theory to the study of symmetries and deformations of the Poisson brackets of hydrodynamic type for d = 2.