Poisson Λ-brackets for Differential–Difference Equations
Poisson Λ-brackets for Differential–Difference Equations
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微分差分方程的泊松 Λ 括号
DOI：
10.1093/imrn/rny242
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发表时间：
2018
影响因子：
1
通讯作者：
M. Wakimoto
中科院分区：
文献类型：
--
作者：
A. Sole;V. Kac;Daniele Valeri;M. Wakimoto
我们引入了乘法泊松$\lambda$-括号的概念，它在哈密顿微分差分方程理论中的作用与通常的泊松$\lambda$-括号在哈密顿偏微分方程（PDE）理论中的作用相同。我们将乘法泊松 $\lambda$-括号分类为一个差异变量，最高可达 5 阶。作为示例，我们演示如何将 Lenard–Magri 方案应用于一对兼容的 1 阶和 2 阶乘法泊松 $\lambda$-括号，以建立 Volterra 链的可积性。
We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential–difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian partial differential equations (PDE). We classify multiplicative Poisson $\lambda$-brackets in one difference variable up to order 5. As an example, we demonstrate how to apply the Lenard–Magri scheme to a compatible pair of multiplicative Poisson $\lambda$-brackets of order 1 and 2, to establish integrability of the Volterra chain.