New Computational Formulae Concerning The Constant γ In The Trace Identity And The Quadratic-form Identity

Fu-Kui Guo

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New Computational Formulae Concerning The Constant γ In The Trace Identity And The Quadratic-form Identity

机译：关于痕量恒等式和二次形式恒等式中的常数γ的新计算公式

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The trace identity and the quadratic-form identity are all simple and powerful tools for establishing Hamiltonian structure of integrable hierarchies of soliton equations, the constant γ contained in the two identities are all to be determined. It has been a left problem to seek for computing formulas on γ, which had been specially proposed by Tu [Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330-81. In this paper, we create an efficient method for obtaining γ by making use of two procedures. First, a few quadratic expressions G(V)'s are discovered from the solvable conditions on Λ, where Λ satisfies the equation [Λ,V]-V_λ=2/λV, whereas, C(V) and γ have the clear, relations. Second, by means of V_x = [U, V], we prove that C(V) is an one-place function with aspect to λ, but not related tox. It follows from the above two steps that the formula γ = λ/2 d/(dλ) ln |C(V)| is obtained. This technique is verified to be feasible and efficient by applying it to a few examples.

机译：迹身份和二次形式身份都是建立孤子方程的可积层的哈密顿结构的简单而强大的工具，两个身份中包含的常数γ都应确定。 Tu [Tu Guizhang。]专门提出的关于计算γ的公式的问题一直是一个左手的问题。痕迹身份是构建可积系统的哈密顿结构的有力工具。 J Math Phys 1989； 30（2）：330-81。在本文中，我们通过使用两个过程创建了一种获取γ的有效方法。首先，从Λ的可解条件中发现一些二次表达式G（V），其中Λ满足方程[Λ，V]-V_λ= 2 /λV，而C（V）和γ清晰，关系。其次，通过V_x = [U，V]，我们证明C（V）是一个单函数，其长宽比与λ无关，但与tox不相关。从以上两个步骤可以得出，公式γ=λ/ 2 d /（dλ）ln | C（V）|获得。通过将其应用于一些示例，可以证明该技术是可行且有效的。

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《Communications in Nonlinear Science and Numerical Simulation》 |2009年第4期|p.1139-1145|共7页
作者
Fu-Kui Guo;
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作者单位

Institute of Mathematics, Information School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China;

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正文语种 eng
中图分类数理科学和化学;
关键词
trace identity; quadratic-form identity; computational formula; lie algebra; loop algebra;

机译：迹身份;二次形式身份;计算公式;李代数;环代数;

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New Computational Formulae Concerning The Constant γ In The Trace Identity And The Quadratic-form Identity

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