Rational Recursion Operators for Integrable Differential-Difference Equations
Rational Recursion Operators for Integrable Differential-Difference Equations
复制标题
可积微分方程的有理递归算子
DOI：
10.1007/s00220-019-03548-8
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发表时间：
2019
影响因子：
2.4
通讯作者：
Carpentier S
中科院分区：
文献类型：
--
作者：
Carpentier S
本文引入了差分算子的预Hamilton对的概念，证明了它们与Nijenhuis算子的关系，并给出了微分差分方程弱非局部逆递归算子存在的一个判据.我们开始严格设置的问题，在斜域的有理数（伪差）运营商的差域与零特征子域的常数和主理想环的矩阵有理（伪差）运营商。特别地，我们给出了有理算子弱非局部的一个判别准则。一个差算子称为预哈密顿算子，如果它的像是差域上关于李括号的李子代数。如果两个预哈密顿算子的任意线性组合是预哈密顿算子，则它们构成预哈密顿算子对。然后，我们表明，一个preHamilton对自然导致Nijenhuis运营商，和Nijenhuis运营商可以表示的preHamilton对。这为检验有理算子是否为Nijenhuis算子提供了一个系统的方法。作为应用，我们构造了一个preHamilton对，从而Nijenhuis递归算子的微分差分方程最近发现的Adler和Postnikov。得到的Nijenhuis算子不是弱非局部的。我们证明了它产生了一个无限的局部交换对称的层次。我们还说明了我们的理论上著名的例子，包括户田，Ablowitz-拉迪克，和Kaup-Newell微分差分方程。
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 ofrational(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.