01006nas a2200097   4500008004100000245007800041210006900119520059600188100001900784856010500803       2015                        en  d00aDispersive deformations of the Hamiltonian structure of Euler's equations0 aDispersive deformations of the Hamiltonian structure of Eulers e3 aEuler'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.1 aCasati, Matteo  uhttps://www.math.sissa.it/publication/dispersive-deformations-hamiltonian-structure-eulers-equations01138nas a2200121   4500008004100000245007900041210006900120260001000189520073700199653002500936100001900961856003600980       2013                        en  d00aOn deformations of multidimensional Poisson brackets of hydrodynamic  type0 adeformations of multidimensional Poisson brackets of hydrodynami  bSISSA3 aThe 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.10aHamiltonian operator1 aCasati, Matteo  uhttp://hdl.handle.net/1963/7235