PreHamiltonian and Hamiltonian operators for differential-difference equations

TL;DR

This paper develops a algebraic theory of rational difference Hamiltonian operators, characterizing their structure and compatibility conditions, and applies it to specific integrable differential-difference equations.

Contribution

It introduces a novel algebraic framework for rational pseudo-difference Hamiltonian operators, including criteria for their Hamiltonian property and compatibility.

Findings

A pseudo-difference Hamiltonian operator can be expressed as a ratio of difference operators with a preHamiltonian numerator.

A skew-symmetric difference operator is Hamiltonian if and only if it is preHamiltonian and meets certain coefficient conditions.

The theory is applied to multi-Hamiltonian structures of specific integrable equations.

Abstract

In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo--difference Hamiltonian operator can be represented as a ratio $AB^{-1}$ of two difference operators with coefficients from a difference field $\mathcal{F}$ where $A$ is preHamiltonian. A difference operator $A$ is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on $\mathcal{F}$. We show that a skew-symmetric difference operator is Hamiltonian if and only if it is preHamiltonian and satisfies simply verifiable conditions on its coefficients. We show that if $H$ is a rational Hamiltonian operator, then to find a second Hamiltonian operator $K$ compatible with $H$ is the same as to find a preHamiltonian pair $A$ and $B$ such that $AB^{-1}H$ is skew-symmetric. We apply our theory to non-trivial multi-Hamiltonian structures of Narita-Itoh-Bogoyavlensky and Adler-Postnikov equations.