Abstract
In this paper we introduce the concept of preHamiltonian pairs of difference operators, demonstrate their connections with Nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential–difference equations. We begin with a rigorous setup of the problem in terms of the skew field of rational (pseudo–difference) operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational (pseudo–difference) operators. In particular, we give a criteria for a rational operator to be weakly nonlocal. A difference operator is called preHamiltonian, if its image is a Lie subalgebra with respect to the Lie bracket on the difference field. Two preHamiltonian operators form a preHamiltonian pair if any 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 systematic 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 and Postnikov. The Nijenhuis operator obtained is not weakly nonlocal. 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.
Highlights
The existence of an infinite hierarchy of commuting symmetries is one of a characteristic property of integrable systems
Symmetries can be generated by recursion operators [1,2], which are often pseudo–differential and map a symmetry to a new symmetry
We introduce and study preHamiltonian pairs of difference operators, their connections with Nijenhuis operators and the existence of weakly nonlocal inverse recursion operators for differential–difference equations
Summary
The existence of an infinite hierarchy of commuting symmetries is one of a characteristic property of integrable systems. We introduce and study preHamiltonian pairs of difference operators, their connections with Nijenhuis operators and the existence of weakly nonlocal inverse recursion operators for differential–difference equations. The Nijenhuis recursion operator of the Volterra chain can be obtained via the Hamiltonian pair, that is, R = H2 H1−1 This decomposition is known as the Lenard scheme used to construct the hierarchies of infinitely many symmetries and cosymmetries. The obtained recursion operator is no longer weakly nonlocal We show that it is Nijenhuis by rewriting it as a rational difference operator and that it generates infinitely many commuting local symmetries. For some integrable differential–difference equations, such as the Ablowitz–Ladik Lattice [19], the recursion operator and its inverse are both weakly nonlocal. To be self-contained, we include “Appendix A”, containing some basic definitions for a unital non-commutative ring
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