Rational Recursion Operators for Integrable Differential–Difference Equations

Abstract

In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin with a rigorous setup of the problem in terms of the skew field Q of rational (pseudo-difference) operators over a difference field F with a zero characteristic subfield of constants k ⊂ F and the principal ideal ring M n (Q) of matrix rational (pseudo-difference) operators. In particular, we give a criteria for a rational operator to be weakly non-local. A difference operator H is called preHamiltonian, if its image is a Lie k-subalgebra with respect the the Lie bracket on F. Two preHamiltonian operators form a preHamiltonian pair if any k-linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematical method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential-difference equation recently discovered by Adler & Postnikov. The Nijenhuis operator obtained is not weakly non-local. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz-Ladik and the Kaup-Newell differential-difference equations.

PreHamiltonian and Hamiltonian operators for differential-difference equations

“…This justifies the necessity to develop a rigorous theory of rational Hamiltonian and recursion operators. In our paper [2] we have extended the results obtained in the differential setting [3] to the difference case. In particular, we have shown that rational recursion operators generating the symmetries of an integrable differential-difference equation must be factorisable as a ratio of two compatible preHamiltonian difference operators.…”

“…The pair of difference operator A and B generates the hierarchy of symmetries of the modified Volterra chain. We have shown in [2] that the difference operators A and B must then form a preHamiltonian pair, that is, any linear combination C = A + λB, λ ∈ k satisfies…”

A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators

“…In this section we revise the formal calculus of variations in the differencedifferential setting as originally laid out by Kupershmidt [12], according to the more modern exposition of [6]. Moreover, we extend the so-called θ formalism to the (difference) local multivector fields.…”

“…It is well known (see, for instance, [6]) that the Volterra chain is an integrable equation admitting a biHamiltonian formulation. Indeed, one can write (4.2) with respect to two compatible Hamiltonian operators…”

“…In (A.4) we have a term of the form (S n−q K (q) ) S −m ∂K(m) ∂u n+m λµ(n − q, −m). (A 6). In (A.5), on the other hand, we have− (S s−q−m K (m) ) S −q ∂K (q) ∂u s θ s−q−m θ −q = (S s−q−m K (m) ) S −q ∂K (q) ∂u s θ −q θ s−q−m .…”