@article {2015,
title = {Dispersive deformations of the Hamiltonian structure of Euler{\textquoteright}s equations},
year = {2015},
abstract = {Euler{\textquoteright}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{\textquoteright}s equation and show that, up to the second order, they are trivial.},
author = {Matteo Casati}
}
@mastersthesis {2015,
title = {Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type},
year = {2015},
note = {161 pages},
school = {SISSA},
abstract = {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 {\textrightarrow} 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{\textendash}Lichnerowicz cohomology. From this computations it
follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.},
keywords = {Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems},
author = {Matteo Casati}
}
@article {2015,
title = {Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets},
year = {2015},
abstract = {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.},
author = {Guido Carlet and Matteo Casati and Sergey Shadrin}
}
@article {2013,
title = {On deformations of multidimensional Poisson brackets of hydrodynamic type},
number = {arXiv:1312.1878;},
year = {2013},
institution = {SISSA},
abstract = {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{\textquoteright}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.},
keywords = {Hamiltonian operator},
url = {http://hdl.handle.net/1963/7235},
author = {Matteo Casati}
}