TY - JOUR
T1 - Dispersive deformations of the Hamiltonian structure of Euler's equations
Y1 - 2015
A1 - Matteo Casati
AB - 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.
U1 - 34700
U2 - Mathematics
U4 - 1
U5 - MAT/07
ER -
TY - THES
T1 - Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type
Y1 - 2015
A1 - Matteo Casati
KW - Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems
AB - 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.
PB - SISSA
N1 - 161 pages
U1 - 34902
U2 - Mathematics
U4 - 1
U5 - MAT/07
ER -
TY - RPRT
T1 - Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets
Y1 - 2015
A1 - Guido Carlet
A1 - Matteo Casati
A1 - Sergey Shadrin
AB - 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.
U1 - 35389
U2 - Mathematics
U4 - 1
U5 - MAT/03
ER -
TY - RPRT
T1 - On deformations of multidimensional Poisson brackets of hydrodynamic type
Y1 - 2013
A1 - Matteo Casati
KW - Hamiltonian operator
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7235
U1 - 7271
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -