# Higher order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type

Casati, Matteo (2018) Higher order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type. Theoretical and Mathematical Physics, 196 (2). pp. 1129-1149. ISSN 0040-5779. (doi:10.1134/S0040577918080032)

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https://doi.org/10.1134/S0040577918080032

## Abstract

The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with twodimensional, two-components Poisson brackets of hydrodynamic type at the third differential degree. This allows us to obtain their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of not equivalent infinitesimal deformation, nor in the third, corresponding to the obstructions to extend such deformations.

Item Type: Article 10.1134/S0040577918080032 Hamiltonian operators, Hydrodynamic Poisson brackets, Poisson Vertex Algebras Q ScienceQ Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Matteo Casati 17 Aug 2018 12:48 UTC 04 Sep 2019 23:00 UTC https://kar.kent.ac.uk/id/eprint/68658 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-2207-4807