%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. %G en %1 34700 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Matteo Casati (mcasati@sissa.it) on 2015-09-09T13:27:08Z No. of bitstreams: 1 defoHD.pdf: 195202 bytes, checksum: 9ee41e73fc37dd79f1659dda2e742b57 (MD5) %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. %I SISSA %G en %U http://hdl.handle.net/1963/7235 %1 7271 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Matteo Casati (mcasati@sissa.it) on 2013-12-09T15:48:14Z No. of bitstreams: 1 1312.1878v1.pdf: 416883 bytes, checksum: d561f4d57afd0607190ba24368e563ae (MD5) %] 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