On deformations of multidimensional Poisson brackets of hydrodynamic type

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 def...

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Published in:arXiv.org 2014-11
Main Author: Casati, Matteo
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Language:English
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Algebra
Brackets
Deformation
Partial differential equations
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description 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 \(d\geq1\) 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 \(d=2\).
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subjects Algebra
Brackets
Deformation
Partial differential equations
title On deformations of multidimensional Poisson brackets of hydrodynamic type
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