%0 Journal Article
%D 2015
%T Dispersive deformations of the Hamiltonian structure of Euler's equations
%A Matteo Casati
%X Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial.
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%1 34700
%2 Mathematics
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%# MAT/07
%$ Submitted by Matteo Casati (mcasati@sissa.it) on 2015-09-09T13:27:08Z
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%0 Thesis
%D 2015
%T Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type
%A Matteo Casati
%K Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems
%X The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical Poisson brackets, defined on an infinite dimensional space of maps Σ → M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson–Lichnerowicz cohomology. From this computations it follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.
%I SISSA
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%1 34902
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%$ Submitted by Matteo Casati (mcasati@sissa.it) on 2015-10-22T07:27:52Z
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%0 Report
%D 2015
%T Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets
%A Guido Carlet
%A Matteo Casati
%A Sergey Shadrin
%X We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D>1. Hence, in contrast with the D=1 case, the deformation theory in the multivariable case is non-trivial.
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%1 35389
%2 Mathematics
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%# MAT/03
%0 Report
%D 2013
%T On deformations of multidimensional Poisson brackets of hydrodynamic type
%A Matteo Casati
%K Hamiltonian operator
%X The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat Hamiltonian partial differential equations. A PVA consist 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 $d\geq 1$ commuting derivations. We call this structure a multidimensional PVA: it is a suitable setting to the study of deformations of the Poisson bracket of hydrodynamic type associated to the Euler's equation of motion of $d$-dimensional incompressible fluids. We prove that for $d=2$ all the first order deformations of such class of Poisson brackets are trivial.
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%U http://hdl.handle.net/1963/7235
%1 7271
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%# MAT/07 FISICA MATEMATICA
%$ Submitted by Matteo Casati (mcasati@sissa.it) on 2013-12-09T15:48:14Z
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%] 1. Introduction
1.1 Poisson Vertex Algebras
1.2 Poisson brackets of hydrodynamic type and their deformations
2. Multidimensional Poisson Vertex Algebras
2.1 Formal map space
2.2 Poisson bivector and Poisson brackets
2.3 Poisson Vertex Algebras
2.4 Proof of Master formula
2.5 Cohomology of Poisson Vertex Algebras
3. Multidimensional Poisson brackets of hydrodynamic type
3.1 Deformation of Lie-Poisson bracket of hydrodynamic type
3.2 Proof of Theorem 5
4. Concluding remarks