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):141252, 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 April 2015
 DOI:
 10.1007/s0022001422192
 arXiv:
 arXiv:1312.1878
 Bibcode:
 2015CMaPh.335..851C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 37K05;
 37K25;
 17B80
 EPrint:
 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