Hirota-Kimura型三维离散系统与变形Lie-Poisson代数
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras
摘要：
最近，Hirota和Kimura提出了一种新的离散化的欧拉顶的几个显着的性质。特别是这种离散化的股票与原来的连续系统的功能，它是一个代数完全可积的双哈密顿系统在三维空间。Hirota-Kimura离散化方案被证明等价于Kahan引入的二次向量场的数值积分方法，Kahan将其应用于二维Lotka-Volterra系统。欧拉顶点自然地用$\mathfrak{so}(3)$李-泊松代数表示。在这里，我们考虑代数上的可积系统，与李-泊松代数对在三维中，如Gümral和Nutku提出的，并构造双有理映射，离散他们根据卡汉和Hirota-Kimura的计划。我们证明了这样得到的映射也是双Hamilton的，具有相容的Poisson括号对，它们是原Lie-Poisson代数的单参数变形，因此它们是完全可积的。为了比较，我们还提出了类似的离散三个双哈密顿系统，具有超越不变量，最后我们分析了所有的Halburd的丢番图可积性标准的观点得到的地图。
英文摘要：
Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the $\mathfrak{so}(3)$ Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.
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