@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}
}
@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}
}
