Dispersive deformations of the Hamiltonian structure of Euler’s equations

Abstract

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.

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

“…First and second order deformations of a scalar reduction of a two-component bracket of hydrodynamic type are trivial, too [7].…”

MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $${\mathcal {W}}$$ W -algebras

“…Furthermore, the notion of a PVA has been extended in [CasPhD,Cas15a] to deal with Hamiltonian operators, or, equivalently, local Poisson brackets, for multidimensional systems of PDEs (namely, PDEs for functions depending on several spatial variables). The notion of multidimensional PVA has been used for studying the theory of symmetries and deformations of the so-called Poisson brackets of hydrodynamic type [DN83], as well as for the local nonlinear brackets associated with 2D Euler's equation [Cas15b].…”