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 of a differential algebra and a bilinear operation called the -bracket. We extend the definition to the class of algebras endowed with 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.
Communications in Mathematical Physics
- Pub Date:
- April 2015
- Mathematics - Differential Geometry;
- Mathematical Physics;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- 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