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

Abstract

The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with twodimensional, two-components Poisson brackets of hydrodynamic type at the third differential degree. This allows us to obtain their corresponding Poisson-Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of not equivalent infinitesimal deformation, nor in the third, corresponding to the obstructions to extend such deformations.

Computing with Hamiltonian operators

“…Total derivatives are distinguished supervector fields; they are defined through a CDIFF function using data that is constructed by CDE. Basically, for each derivative symbol (like u 2tx) in the system the coefficient of the corresponding derivative ∂/∂u 2tx is computed by increasing the multiindex by one in the direction of the corresponding independent variable, and the results are summed up to make (14). The total derivative of a superfunction phi is invoked by td(phi,x); or td(phi,t,x,2);, for example.…”

“…CDE has this capability, the interested Reader can find one simple example (provided by Casati) in the CDE manual. More non-trivial examples, with applications to Hamiltonian and bi-Hamiltonian cohomology, are computed (also by CDE) in the recent paper [14].…”