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A Higher Dimensional loop algebra and Integrable Couplings System of Evolution Equations Hierarchy

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Journal of Shanghai University(English Edition) 2005年第3期9卷 201-205页

作者：夏铁成于发军陈登远 Department of Mathematics College of Sciences Shanghai University Shanghai 200444 P.R. China Department of Mathematics Bohai University Jinzhou 121000 P.R. China Department of Mathematics College of Sciences Shanghai University Shanghai 200444 P.R. China

An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loop algebra [AKG～], the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.

关键词： Lie algebra integrable couplings system loop algebra NLS-MKdV equations hierarchy.

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A subalgebra of loop algebra A2^～ and its applications

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Chinese Physics B 2004年第2期13卷 132-138页

作者：张玉峰谭汉华郭福奎 InstituteofMathematics InformationSchoolShandongUniversityofScienceandTechnologyTaian271019China DepartmentofComputerScience HongKongBaptistUniversityKowloonTongHongKongChina

A subalgebra of loop algebra "s_2 and its expanding loop algebra \overline{G} are constructed. It follows that both resulting integrable Hamiltonian hierarchies are obtained. As a reduction case of the first hierarchy, a generalized nonlinear coupled Schr"sdinger equation, the standard heat-conduction and a formalism of the well known Ablowitz, Kaup, Newell and Segur hierarchy are given, respectively. As a reduction case of the second hierarchy, the nonlinear Schr"sdinger and modified Korteweg de Vries hierarchy and a new integrable system are presented. Especially, a coupled generalized Burgers equation is generated.

关键词： loop algebra integrable hierarchy Hamiltonian structure

An integrable Hamiltonian hierarchy, a highdimensional loop algebra and associated integrable coupling system

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Chinese Physics B 2003年第11期12卷 1194-1201页

作者：张玉峰 InstituteofMathematics InformationSchoolShandongUniversityofScienceandTechnologyTaian271019China

A subalgebra of loop algebra A_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu"s scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schrodinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra A_2 into loop algebra A_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

关键词： loop algebra integrable system Hamiltonian structure constrained flow

Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation

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Science China Mathematics 2021年第6期64卷 1181-1200页

作者： Fulin Chen Naihuan Jing Fei Kong Shaobin Tan School of Mathematical Sciences Xiamen UniversityXiamen 361005China Department of Mathematics North Carolina State UniversityRaleighNC 27695USA School of Mathematics and Statistics Hunan Normal UniversityChangsha 410006China

Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g with respect toμ,which provides a Moody-Rao-Yokonuma presentation for the algebraˆg[μ]whenμis non-transitive,and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.

关键词： Moody-Rao-Yokonuma presentation loop algebra universal central extension extended affine Lie algebra

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一类非线性演化方程族的可积耦合

高等学校计算数学学报 2005年第1期27卷 46-52页

作者：张玉峰许曰才闫庆友华北电力大学(北京)工商管理学院北京 102206 大连 116029 山东科技大学成人教育学院泰安 271019 辽宁师范大学数学学院

1　引言众所周知,寻求可积系统是孤立子理论中的一项重要研究课题.如何寻求已知可积系统的扩展可积系统是寻求可积系理论中的一个重要方面,对此我们已经做了一些工作 [2-4],本文中我们考虑了文献[5]提出的一个Lax对,我们称其为屠-Lax... 详细信息

1　引言众所周知,寻求可积系统是孤立子理论中的一项重要研究课题.如何寻求已知可积系统的扩展可积系统是寻求可积系理论中的一个重要方面,对此我们已经做了一些工作 [2-4],本文中我们考虑了文献[5]提出的一个Lax对,我们称其为屠-Lax对,即这里β1,β2为常数,且β1≠β2;p1,p2,q,r为位势函数;δ1n,δ2n为任意常数或函数. 由文献[5]知,(1)的相容性导出一类非线性演化方程族 p1t-δ1nx,p2t=δ2nx,qt=βbn+1+(δ1n-δ2n)q,rt=βcn+1+(δ2n-δ1n)r. (2)

关键词： integrable coupling loop algebra isospectral problem.

The extended trace identity and its application

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Chinese Physics B 2007年第3期16卷 611-620页

作者：姚玉芹陈登远 Department of Mathematics Shanghai University Shanghai 200444 China

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable sys- tems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de-Vries （KdV） hierarchy, the multi-component Ablowitz-Kaup-Newell-Segur （M-AKNS） hierarchy, the multi-component Ablowitz-Kaup-Newell-Segur Kaup-Newell （M-AKNS-KN） hierarchy and a new multi-component integrable hierarchy separately.

关键词： loop algebra Killing form trace identity Hamiltonian structure

Two types of loop algebras and their expanding Lax integrable models

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Chinese Physics B 2007年第3期16卷 588-594页

作者：岳超张玉峰魏媛 School of Information Engineering Taishan Medical University Taian 271016 China School of Information Science and Engineering Shandong University of Science and Technology Qingdao 266510 China

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation Ut-Vx ＋[U, V] = 0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator ^~J is presented by constructing a subalgebra ^~G of the loop algebra -^~A2. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their （2＋1）-dimensional forms are derived from a （2＋1）-dimensional zero-curvature equation.

关键词： zero-curvature equation integrable hierarchy loop algebra

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

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Chinese Physics B 2007年第3期16卷 605-610页

作者：夏铁成尤福财 Department of Mathematics Shanghai University Shanghai 200444 China Department of Mathematics Tianjin University Tianjin 300072 China

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra ~FM of the loop algebra ~X is presented. Based on the ~FM, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

关键词： loop algebra zero curvature equation multi-component Dirac equation hierarchy multi-component integrable couplings system

An integrable Hamiltonian hierarchy and associated integrable couplings system

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Chinese Physics B 2007年第9期16卷 2493-2497页

作者：陈晓红夏铁成朱连成 School of Science University of Science and Technology Liaoning Liaoning 114044 China Department of Mathematics Shanghai University Shanghai 200444 China School of Electronic and Information Engineering University of Science and Technology Liaoning Liaoning 114044 China

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new intcgrablc system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary sy... 详细信息

关键词： integrable system Hamiltonian structure loop algebra