Study of Solving a Toda Dynamic System with Loop Algebra

ZHU Qiao; YANG Zhan-Ying; SHI Kang-Jie; WEN Jun-Qing

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2024年10月30日

Chinese Physics C> 2006, Vol. 30> Issue(9) : 838-843

Study of Solving a Toda Dynamic System with Loop Algebra

Institute of Science, Shaanxi University of Science & Technology, Xianyang 712081, China2 Department of Physics, Northwest University, Xi'an 710069, China3 Institute of Modern Physics, Northwest University, Xi'an 710069, China4 Institute of Science, Xi'an Shiyou University, Xi'an 710065, China

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Abstract：
In this paper, we construct a Toda system with Loop algebra, and prove that the Lax equation L=[L,M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L=L⁺+M, and L⁺ is a quasi-upper triangular matrix of loop algebra. In order to check our result, we exactly solve a R-H problem under a given initial condition as an example.

References

[1]

. Toda M J. Phys. Soc. Japan, 1967, 23: 501-5062. Castro-Alvaredo O A, Fring A. Applications of QuantumIntegrable Systems. arXiv:hep-th/03011023. JIANG Jin-Huan, LI Ai-Min, LI Zi-Ping. HEP NP, 2003,27(6): 489-492(in Chinese)(江金环，李爱民，李子平.高能物理与核物理，2003, 27(6):489-492)4. YANG Zhan-Ying, ZHEN Yi. HEP NP, 2000, 24(6):484-489(in Chinese)(杨战营，甄翼.高能物理与核物理，2000, 24(6): 484-489)5. YANG Zhan-Ying, ZHAO Liu, ZHEN Yi. HEP NP, 2000,24(4): 1-7(in Chinese)(杨战营，赵柳，甄翼.高能物理与核物理，2000, 24(4): 1-7)6. Nirov K S, Razumov A V. Higher Symmetries of TodaEquations. arXiv:hep-th/02101367. Toda M. Prog. Theor. Phys. Suppl., 1970, 45: 174-2008. Moser J. Adv. Math., 1975, 16: 197-220; Birkhauser.Basel, Prog. in Math., 1980, 8: 233-2899. Ruijsenaars S N M. Commun. Math. Phys., 1987, 110:191-21310. LIU Wang-Yun, YANG Zhan-Ying, ZHAO Liu. HEP NP,2004, 28(4): 359-364(in Chinese)(朱桥，杨战营，石康杰.高能物理与核物理，2005, 29(1): 19-22)11. ZHAO Liu, LIUWang-Yun. Commun. Theore. Phys., 2004,42: 9-1812. ZHU Qiao, YANG Zhan-Ying, SHI Kang-Jie. HEP NP,2005, 29(1): 19-2213. HUANG Lie-De. Advances in Mathematics, 1990, 19(2):265-283(in Chinese)(黄烈德.数学进展，1990, 19(2): 265-283)

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ZHU Qiao, YANG Zhan-Ying, SHI Kang-Jie and WEN Jun-Qing. Study of Solving a Toda Dynamic System with Loop Algebra[J]. Chinese Physics C, 2006, 30(9): 838-843.

ZHU Qiao, YANG Zhan-Ying, SHI Kang-Jie and WEN Jun-Qing. Study of Solving a Toda Dynamic System with Loop Algebra[J]. Chinese Physics C, 2006, 30(9): 838-843. shu

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Received: 2005-12-17
Revised: 2006-04-22

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沈阳化工大学材料科学与工程学院沈阳 110142

Study of Solving a Toda Dynamic System with Loop Algebra

Corresponding author: ZHU Qiao,

Institute of Science, Shaanxi University of Science & Technology, Xianyang 712081, China2 Department of Physics, Northwest University, Xi'an 710069, China3 Institute of Modern Physics, Northwest University, Xi'an 710069, China4 Institute of Science, Xi'an Shiyou University, Xi'an 710065, China

Received Date: 2005-12-17
Accepted Date: 2006-04-22
Available Online: 2006-09-05

Abstract: In this paper, we construct a Toda system with Loop algebra, and prove that the Lax equation L=[L,M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L=L⁺+M, and L⁺ is a quasi-upper triangular matrix of loop algebra. In order to check our result, we exactly solve a R-H problem under a given initial condition as an example.