两个loop代数及其应用

梁凤鸣

泰山学院学报编辑部,山东泰安271021

两个loop代数及其应用

梁凤鸣

泰山学院学报编辑部,山东泰安271021

基于两个loop代数,利用(2+1)-维零曲率方程分别得到了(2+1)-维超AKNS族和超Tu族.

loop代数;屠格式;(2+1)-维零曲率方程

寻找尽可能多的可积孤立子方程族是孤立子理论研究中的一项重要而有趣的课题,目前流行的方法是屠格式,人们利用屠格式获得了许多有重要物理意义的可积方程族,如AKNS,KN,BPT族等[1-3];随着可积系统研究的不断深入,(2+1)-维可积和超可积系统引起了研究兴趣[4-9].在文献[10]中周子翔教授引入了如下的Lax对

其相容性条件为

1 (2+1)维超AKNS方程族

在文献[6]中,马文秀教授给出了如下的loop代数B(0,1)

其中E1，E2，E3是偶元,E4,E5是奇元,符号[,]和[,]分别表示交换子和反交换子.相应的loop代数B~(0,1)为

2 (2+1)维超Tu族

作E1,E2,…,E5的线性组合,则可得如下的loop代数B1(0,1)

[1]Zhang Tu-Gui.The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J. Math.Phys.,1989,30(2):330-338[2]郭福奎.Loop代数的子代数与可积Hamilton方程族[J].数学物理学报,1999,19(5):507-512

[3]Zhang Yu-feng.A general Boite-Pempinelli-Tu hierarchy and its bi-Hamiltonian structure[J].Phys.Lett.A,2003,317(3):280-286

[4]Zhang Yu-feng,Guo Xiu-rong.A(2+1)-dimensional integrable hierarchy and its extending integrable model[J]. Chaos,Solitons and Fractals,2006,27:555-559

[5]Zhou Zi-xiang.Finite dimensional Hamiltonians andalmaost-periodic solutions for 2+1-dimensional three-wave equations [J].J.Phys.Jpn.,2002,71(8):1857-1863

[6]Ma Wen-xiu,He Jing-song,Qin Zhen-yun.A supertrace identity and its applications to superintegrable systems[J]. Journal of mathematical physics,2008,49:033511

[7]Wang Xin-zeng,Liu Xi-kui.Two types of Lie super-algebra for the super-integrable Tu-hierarchy and its super-Hamiltonian structure[J].Commun.Nonlinear Sci.Numer.Simul.,2010,15:2044-2049

[8]Zhang Yu-feng,Gao Jing.Wang Guang-ming.Two(2+1)-dimensional hierarchies of evolution equations and their Hamiltonian structures[J].Appl.Math.Comput.,2014,243:601-606

[9]Zhang Yu-feng,Rui Wen-juan.On generating(2+1)-dimensional hierarchies of evolution equations[J].Commun Nonlinear Sci.Numer.Simul.2014,19:3454-3461

[10]Zhou Zi-xiang.Finite dimensional Hamiltonians and almost-periodic solutions for 2+1-dimensional three-wave equations[J].J.Phys.Jpn.2002,71(8):1857-1863

Two Super-loop Algebras and Its Application

LIANG Feng-ming
Editorial Department of Journal Taishan University,Taian 271021,China

This paper respectively obtained the(2+1)-dimensional super-AKNS hierarchy and(2+1)-dimensional super-Tu hierarchy with(2+1)-dimensional zero curvature equation based on two super-loop algebras.

Super-loop algebra;Tu scheme;(2+1)-dimensional zero curvature equation

O175.29文献标示码:A

1000-2324(2015)03-0457-04

2013-07-04

2013-07-24

山东省自然科学基金项目(ZR2012AL03)

梁凤鸣(1979-),女,编辑,主要从事期刊编辑研究工作.E-mail:liang-fengming@163.com