中国物理B ›› 2003, Vol. 12 ›› Issue (11): 1194-1201.doi: 10.1088/1009-1963/12/11/302

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An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

张玉峰   

  1. Institute of Mathematics, Information School, Shandong University of Science and Technology, Taian 271019, China; Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
  • 收稿日期:2003-03-14 修回日期:2003-06-05 出版日期:2003-11-16 发布日期:2005-03-16

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

Zhang Yu-Feng (张玉峰)ab   

  1. a Institute of Mathematics, Information School, Shandong University of Science and Technology, Taian 271019, China; b Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
  • Received:2003-03-14 Revised:2003-06-05 Online:2003-11-16 Published:2005-03-16

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摘要： A subalgebra of loop algebra $\tilde{A}_2$ 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 Schr?dinger 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 $\tilde{A}_2$ into loop algebra $\tilde{A}_1$. 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.

Abstract: A subalgebra of loop algebra $\tilde{A}_2$ 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 Schr?dinger 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 $\tilde{A}_2$ into loop algebra $\tilde{A}_1$. 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.

Key words: loop algebra, integrable system, Hamiltonian structure, constrained flow

中图分类号:  (Integrable systems)

  • 02.30.Ik
03.65.Ge (Solutions of wave equations: bound states) 03.65.Fd (Algebraic methods)