Higher order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type

TL;DR

This paper investigates the algebraic structure and deformations of multidimensional Poisson brackets of hydrodynamic type, computing their cohomology to understand possible higher order dispersive deformations.

Contribution

It computes the Poisson-Lichnerowicz cohomology for 2D, 2-component hydrodynamic Poisson brackets, revealing the structure of their deformations and obstructions.

Findings

Cohomology is non-trivial in second and third groups.

Identifies classes of non-equivalent infinitesimal deformations.

Provides a foundation for understanding higher order dispersive deformations.

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 two-dimensional, 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 obstruction to extend such deformations