Multi-component Hamiltonian difference operators

TL;DR

This paper classifies low order local Hamiltonian operators for two-component differential-difference equations and analyzes their Poisson cohomology, revealing insights into their deformation theory and bi-Hamiltonian structures.

Contribution

It extends classification results to degenerate cases and computes Poisson cohomology for a key two-component operator, advancing understanding of integrable systems.

Findings

Classified low order Hamiltonian operators including degenerate cases

Computed Poisson cohomology for a (-1,1)-order operator

Demonstrated implications for bi-Hamiltonian structures in integrable systems

Abstract

In this paper we study local Hamiltonian operators for multi-component evolutionary differential-difference equations. We address two main problems: the first one is the classification of low order operators for the two-component case. On the one hand, this extends the previously known results in the scalar case; on the other hand, our results include the degenerate cases, going beyond the foundational investigation conducted by Dubrovin. The second problem is the study and the computation of the Poisson cohomology for a two-component (-1,1)-order Hamiltonian operator with degenerate leading term appearing in many integrable differential-difference systems, notably the Toda lattice. The study of its Poisson cohomology sheds light on its deformation theory and the structure of the bi-Hamiltonian pairs where it is included in, as we demonstrate in a series of examples.