2n维空间中的广义自对偶Yang-Mills方程的达布变换

数学物理学报 ›› 2015, Vol. 35 ›› Issue (3): 478-486.

2n维空间中的广义自对偶Yang-Mills方程的达布变换

沈守枫¹, 于水猛², 李春霞³, 金永阳¹

1. 浙江工业大学应用数学系杭州 310023;
2. 江南大学理学院江苏无锡 214122;
3. 首都师范大学数学科学学院北京 100048

收稿日期:2014-04-11 修回日期:2014-10-28 出版日期:2015-06-25 发布日期:2015-06-25
作者简介:沈守枫, mathssf@zjut.edu.cn
基金资助:
国家自然科学基金(11371323, 11271266)资助

Darboux Transformation for A Generalized Self-Dual Yang-Mills Equation in 2n Dimensions

Shen Shoufeng¹, Yu Shuimeng², Li Chunxia³, Jin Yongyang¹

1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023;
2. School of Sciences, Jiangnan University, Jiangsu Wuxi 214122;
3. School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received:2014-04-11 Revised:2014-10-28 Online:2015-06-25 Published:2015-06-25

摘要/Abstract

摘要：

从带负幂次谱参数的谱问题出发,构造了一类广义自对偶Yang-Mills方程. 这类方程包括若干著名的Lax可积方程, 如Takasaki情形、Belavin-Zakharov情形、Ablowitz-Chakravarty-Takhtajan情形和Ma情形. 进而建立了这类方程的达布变换的精确表达式.

关键词: 达布变换, 自对偶Yang-Mills方程, Lax可积, 谱问题

Abstract:

A generalized self-dual Yang-Mills equation with negative powers of the spectral parameter is proposed by a set of spectral problems. It contains some well-known Lax integrable equations such the Takasaki case, the Belavin-Zakharov case, the Ablowitz-Chakravarty-Takhtajan case and the Ma case. The explicit formulation of Darboux transformation is established for this equation.

Key words: Darboux transformation, Self-dual Yang-Mills equation, Lax integrable, Spectral problem

中图分类号:

沈守枫, 于水猛, 李春霞, 金永阳. 2n维空间中的广义自对偶Yang-Mills方程的达布变换[J]. 数学物理学报, 2015, 35(3): 478-486.

Shen Shoufeng, Yu Shuimeng, Li Chunxia, Jin Yongyang. Darboux Transformation for A Generalized Self-Dual Yang-Mills Equation in 2n Dimensions[J]. Acta mathematica scientia,Series A, 2015, 35(3): 478-486.

参考文献

[1] Yang C N. Condition of selfduality for SU(2) gauge fields on Euclidean four-dimensional space. Phys Rev Lett, 1977, 38: 1377-1379
[2] Corrigan E R, Fairlie D B, Yates R G, Goddard R. The construction of self-dual solutions to SU(2) gauge theory. Commun Math Phys, 1978, 58: 223-240
[3] Chau L L, Prasard M K, Sinha A. Some aspects of the linear systems for self-dual Yang-Mills fields. Phys Rev D, 1981, 24: 1574-1580
[4] Takasaki K. A new approach to the self-dual Yank-Mills equations. Commun Math Phys, 1984, 94: 35-59
[5] Ward R S. Integrable and solvable systems and relations among them. Phil Trans Roy Soc London A, 1985, 315: 451-457
[6] Zhou Z X. On the Darboux transformation for 1+2-dimensional equations. Lett Math Phys, 1988, 16: 9-17
[7] Ablowitz M J, Chakrayarty S, Takhtajan L A. A self-dual Yang-Mills hierarchy and its reduction to integrable systems in 1+1 and 2+1 dimensions. Commun Math Phys, 1993, 159: 289-314
[8] Gu C H, Zhou Z X. On Darboux transformations for soliton equations in high-dimensional spacetime. Lett Math Phys, 1994, 32: 1-10
[9] Gu C H. Integrable evolution systems based on generalized self-dual Yang-Mills equations and their soliton-like solutions. Lett Math Phys, 1995, 35: 61-74
[10] Gu C H. Generalized self-dual Yang-Mills flows, explicit solutions and reductions. Acta Appl Math, 1995, 39: 349-360
[11] Ma W X. Darboux transformations for a Lax integrable system in 2n dimensions. Lett Math Phys, 1997, 39: 33-49
[12] Ustinov N V. The reduced self-dual Yang-Mills equation, binary and infinitesimal Darboux transformations. J Math Phys, 1998, 39: 976-985
[13] Ward R S. Two integrable systems related to hyperbolic monopoles. Asian J Math, 1999, 3: 325-332
[14] Nimmo J J C, Gilson C R, Ohta Y. Applications of Darboux transformations to the selfdual Yang-Mills equations. Theor Math Phys, 2000, 122: 239-246
[15] Zhou Z X. Solutions of the Yang-Mills-Higgs equations in 2+1 dimensional anti-de Sitter space-time. J Math Phys, 2001, 42: 1085-1099
[16] Gu C H. Darboux transformation and solitons of Yang-Mills-Higgs equations in R^{2, 1}. Science in China Ser A, 2002, 45: 706-715
[17] Saleem U, Hassan M, Siddiq M. Non-local continuity equations and binary Darboux transformation of noncommutative (anti) self-dual Yang-Mills equations. J Phys A: Math Theor, 2007, 40: 5205-5217
[18] Gilson C R, Hamanaka M, Nimmo J J C. Bäcklund transformations and the Atiyah-Ward ansatz for non-commutative anti-self-dual Yang-ills equations. Proc R Soc A, 2009, 465: 2613-2632
[19] Gilson C R, Hamanaka M, Nimmo J J C. Bäklund transformations for noncommutative anti-self-dual Yang-Mills equations. Glasgow Math J, 2009, 51: 83-93
[20] Tchrakian D H. Notes on Yang-Mills-Higgs monopoles and dyons on R^D, and Chern-Simons-Higgs solitons on R^D-2: dimensional reduction of Chern-Pontryagin densities. J Phys A: Math Theor, 2011, 44: 343001, 53pages
[21] Hamanaka M. Noncommutative solitons and quasideterminants. Phys Scr, 2014, 89: 038006, 11pages