On deformations of multidimensional Poisson brackets of hydrodynamic type

Abstract

The theory of Poisson Vertex Algebras (PVAs) 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 $d\ge 1$ commuting derivations. We call this structure a \emph{multidimensional PVA}: it is a suitable setting to study Hamiltonian PDEs with $d$ spatial dimensions. We apply this theory to the study of deformations of the Poisson brackets of hydrodynamic type for $d=2$.

Revision with shorter exposition of the content of Sec 2 and new results about first cohomology groups. 50 pages. Reference and equation numbers fixed with respect to version 3

References in corpus (1)

• Hamiltonian operators of Dubrovin-Novikov type in 2D

Cited by in corpus (6)

• Hamiltonian operators of Dubrovin-Novikov type in 2D
• Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets
• Normal forms of dispersive scalar Poisson brackets with two independent variables
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras
• Dispersive deformations of the Hamiltonian structure of Euler's equations
• A construction of Multidimensional Dubrovin-Novikov Brackets