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Multi-component Hamiltonian difference operators

Matteo Casati, Daniele Valeri

TL;DR

This work advances the theory of multi-component difference Hamiltonian operators by deriving necessary and sufficient conditions for (-1,1)-order operators to be Hamiltonian in the two-component setting and by establishing normal forms under point transformations, including degenerate leading terms. It also develops and applies the Poisson cohomology framework to analyze deformation and bi-Hamiltonian structures, demonstrating that for the Toda lattice the cohomology is trivial beyond degree two and that higher-order deformations are Miura-trivial. Using these cohomological results, the authors construct explicit local two-component bi-Hamiltonian pairs for several integrable differential-difference lattices (Toda, Bruschi-Ragnisco, Volterra, relativistic variants), showing how higher Hamiltonian structures arise as coboundaries of the base structure. The findings illuminate the geometric structure of bi-Hamiltonian hierarchies in discrete settings and suggest directions for extending the theory to nonlocal operators and higher-component systems with richer deformation spaces, reinforcing the central role of (-1,1)-order operators in differential-difference integrability.

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.

Multi-component Hamiltonian difference operators

TL;DR

This work advances the theory of multi-component difference Hamiltonian operators by deriving necessary and sufficient conditions for (-1,1)-order operators to be Hamiltonian in the two-component setting and by establishing normal forms under point transformations, including degenerate leading terms. It also develops and applies the Poisson cohomology framework to analyze deformation and bi-Hamiltonian structures, demonstrating that for the Toda lattice the cohomology is trivial beyond degree two and that higher-order deformations are Miura-trivial. Using these cohomological results, the authors construct explicit local two-component bi-Hamiltonian pairs for several integrable differential-difference lattices (Toda, Bruschi-Ragnisco, Volterra, relativistic variants), showing how higher Hamiltonian structures arise as coboundaries of the base structure. The findings illuminate the geometric structure of bi-Hamiltonian hierarchies in discrete settings and suggest directions for extending the theory to nonlocal operators and higher-component systems with richer deformation spaces, reinforcing the central role of (-1,1)-order operators in differential-difference integrability.

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.

Paper Structure

This paper contains 23 sections, 14 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Multiplicative Poisson vertex algebras and poly-vectors
  3. Algebras of difference functions and vector fields
  4. Difference operators and multiplicative Poisson vertex algebras
  5. Local poly-vectors and the theta formalism
  6. Homomorphisms of algebras of difference functions and point transformations
  7. Hamiltonian matrix difference operators
  8. Normal forms for l=2 difference Hamiltonian operators
  9. The Hamiltonian difference operator corresponding to Aff(R)
  10. The Hamiltonian difference operator corresponding to the 2-dimensional Abelian lie group
  11. The case when the matrix A is degenerate
  12. The Poisson cohomology
  13. The Poisson cohomology of a difference Hamiltonian structure
  14. The Poisson cohomology of an ultralocal structure
  15. The Poisson cohomology of H0
  16. ...and 8 more sections

Key Result

Proposition 6

Let $K$ be a Hamiltonian difference operator of the form eq:diffop-1st. If $A$ is nondegenerate, then for every $i,j=1,\dots,\ell$ and denoting $u_q=(u^1_q,\ldots,u^\ell_q)$.

Theorems & Definitions (43)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • Remark 8
  • ...and 33 more