超Dirac方程族的对称及其Lie代数
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摘要
文中讨论了超Dirac方程族和超AKNS方程族的对称及其Lie代数结构。首先,我们证明了方程族的递推算子φ是一个遗传强对称。基于此,找到了超Dirac方程族的2N2个对称,并建立了它们的无穷维Lie代数结构。通过与一般的Dirac方程族的对称及其Lie代数结构作比较,我们发现,超变量的的引入没有影响其结果。进一步研究发现,超AKNS方程族和一般的AKNS方程族也具有相同的对称及其Lie代数结构形式。
In this paper, the symmetries and its Lie algebra structure of the super Dirac hierarchy and super AKNS hierarchy are discussed. First, we find that the recursion operator0of the hierarchy is a strong hereditary symmetry. From this,2N2families of symmetries and their infinite-dimensional Lie algebra structure are proposed. By contrast the symmetries and Lie algebra structure with these of general Dirac hierarchy, we find that the intro-ducing of super variables have no influence. Furthermore, the super AKNS hierarchy and general AKNS hierarchy also have the same form of symmetries and Lie algebra structure.
In this paper, the symmetries and its Lie algebra structure of the super Dirac hierarchy and super AKNS hierarchy are discussed. First, we find that the recursion operator0of the hierarchy is a strong hereditary symmetry. From this,2N2families of symmetries and their infinite-dimensional Lie algebra structure are proposed. By contrast the symmetries and Lie algebra structure with these of general Dirac hierarchy, we find that the intro-ducing of super variables have no influence. Furthermore, the super AKNS hierarchy and general AKNS hierarchy also have the same form of symmetries and Lie algebra structure.
引文
[1]Olver, P.J, Applicationsof Lie Groups to Differential Equations, Springer-Verlag, New York, (1986), Chapter 5.
[2]Yao Yu-qin, Liu Yu-qing, Ji Jie, and Chen Deng-yuan, The Second mKdV Equation and its Hereditary Symmetry and Hamiltonian Structure, Commun.Theor.Phys. (Beijing, China)47(2007)pp.22-24.
[3]田畴,MKdV方程的两组对称及其几何意义,高校应用数学学报,(1989),No.2,Vo1.4.
[4]李忠定,Kaup-Newell族的换位表示,数学物理学报,12(1992),1,68-74.
[5]Zhu Guocheng and Ji Xiaoda, Symmetries and Lie Algebra for the New Hierarchy of Heisenberg Equations, Journal of China University of Science and Technology, vol.25, No.2, (1995),119-126.
[6]田畴,Burgers方程的新的强对称,对称和李代数,中国科学,(1987).
[7]Xu Bin, Recursion Operator, Lie Algebra and conservation Laws of the Burgers Equation, Journal of Liaocheng University(Nat.Sci.), Vol.23, No.4(2010).
[8]Fuchssteiner B.Integrable Nonlinear Evolution Eguations with Time-dependeng Coeffi-cients.J Math Phys.(1993),5140-5158.
[9]Chen Deng-yuan, Zhu Guo-chen, and Li Yi-shen, The Symmetry and its Lie Algebra for the Matrix Evolution Equation of KdV Type, Acta Mathematicae Applicatae Sinica, Vol.13.No.3(1990).
[10]Yi Cheng and Yi-shen Li, Symmetries and constants of notion for new AKNS hierarchies, J.Phys.A:Math.Gen.20(1987) 1951-1959.
[11]陈登远,朱国城,李翊神,AKNS型矩阵发展方程的新对称及其Lie代数,数学年刊,12A：1(1991),33-42.
[12]Li Yi-shen and Zhu Guo-cheng, New Set of Symmetries of the Integrable Equations, Lie Algebra and Non-isospectral Evolution Equations:Ⅱ.AKNS System, J.Phys.A:Math. Gen.19(1986) 3713-3725.
[13]王宝勤,张飞军,源于KdV方程的延拓结构的方程与对称,新疆师范大学学报(自然科学版),第17卷第2期,(1998),1-7.
[14]斯仁道尔吉,组合KdV方程的强对称、对称及其Lie代数,内蒙古师大学报(自然科学汉文版),第3期,数学增刊,(1992),51-57.
[15]陈守婷,半离散AKNS系统的对称与代数结构,博士学位论文,上海大学,(2011).
[16]袁洪芬,超空间上Dirac型方程解的性质,博士学位论文,河北师范大学,(2012).
[17]谷超豪等著,孤立子理论与应用,浙江科学出版社,(1990)216-267.
[18]陈登远编著,孤子引论.-北京：科学出版社,(2006)46-68,187-227.
[19]李翊神编著,孤子与可积系统.上海：上海科技教育出版社,(1999.12)1-40,111-151.
[20]朱国城,关于强对称和遗传对称算子性质的几点注记,科学通报,第2期,(1986).
[21]李翊神,朱国城,可积方程的新的对称、李代数及谱可变演化方程(Ⅰ),中国科学,A辑,第3期,(1987),235-241.
[22]朱国城,李翊神,可积方程的新对称、李代数与谱可变发展方程(Ⅲ),科学通报,第24期,(1986),1845-1849.
[23]马文秀,一类Hamilton算子、遗传对称及可积系,高校应用数学学报,第8卷第1期,(1993),28-35.
[24]王李勇,胡星标,一些非线性演化方程族的新对称,应用数学与计算数学学报,第2卷第2期,(1988),74-82.
[25]Wen-Xiu Ma and Kam-Shun Li, Virasoro Symmetry Algebra of Dirac Soliton Hierarchy, Inverse problems,12(1996)L25-L31.
[26]Jing Yu, Jing-song He, Wen-Xiu Ma, and Yi Cheng, The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac Systems, (2009).
[27]Sirendaoreji, Hamiltonian Structures for the Constrained Flows of the Dirac Hierar-chy, Journal of Inner Monggolia Normal University(Natural Science Edition), Vol.34, No.3(2005).
[28]Yong Fang, Yijun Hou, Huanhe, Jianhai Xue, New Super-hamiltonian Structures of Super-Dirac Hierarchy, Ann.of Diff.Eqs.28:2(2012),164-169.
[29]马文秀,向量场的生成子与发展方程的时间依赖对称,中国科学,第2期,(1991),129-139.
[30]王鸿业,一类系数依赖时间的非线性演化方程的对称及其李代数,郑州大学学报(自然科学版),第27卷第4期,(1995),7-11.
[31]Xin-Zeng Wang and Huan-He Dong, A Lie Superalgebra and Corresponding Hierarchy of Evolution Equations, Modern Physics Letters B, Vol.23, No.28(2009)3387-3396.
[32]Zhu Guocheng, Chen Dengyuan, Zeng Yunbo, A Class Matrix Evolution Equations, Chin.Ann.of Math.3 (1) (1982).
[33]杜殿楼,Lie-Poisson框架下的Dirac-Bargmann系统的可积性,河南科学,第23卷第4期,(2005),472-475.
[34]陶司兴,李超代数与非线性演化方程族的研究,博士学位论文,上海大学,(2011).
[35]连增菊,非线性方程中的对称和对称约化的新进展,硕士学位论文,宁波大学,(2004).
[2]Yao Yu-qin, Liu Yu-qing, Ji Jie, and Chen Deng-yuan, The Second mKdV Equation and its Hereditary Symmetry and Hamiltonian Structure, Commun.Theor.Phys. (Beijing, China)47(2007)pp.22-24.
[3]田畴,MKdV方程的两组对称及其几何意义,高校应用数学学报,(1989),No.2,Vo1.4.
[4]李忠定,Kaup-Newell族的换位表示,数学物理学报,12(1992),1,68-74.
[5]Zhu Guocheng and Ji Xiaoda, Symmetries and Lie Algebra for the New Hierarchy of Heisenberg Equations, Journal of China University of Science and Technology, vol.25, No.2, (1995),119-126.
[6]田畴,Burgers方程的新的强对称,对称和李代数,中国科学,(1987).
[7]Xu Bin, Recursion Operator, Lie Algebra and conservation Laws of the Burgers Equation, Journal of Liaocheng University(Nat.Sci.), Vol.23, No.4(2010).
[8]Fuchssteiner B.Integrable Nonlinear Evolution Eguations with Time-dependeng Coeffi-cients.J Math Phys.(1993),5140-5158.
[9]Chen Deng-yuan, Zhu Guo-chen, and Li Yi-shen, The Symmetry and its Lie Algebra for the Matrix Evolution Equation of KdV Type, Acta Mathematicae Applicatae Sinica, Vol.13.No.3(1990).
[10]Yi Cheng and Yi-shen Li, Symmetries and constants of notion for new AKNS hierarchies, J.Phys.A:Math.Gen.20(1987) 1951-1959.
[11]陈登远,朱国城,李翊神,AKNS型矩阵发展方程的新对称及其Lie代数,数学年刊,12A：1(1991),33-42.
[12]Li Yi-shen and Zhu Guo-cheng, New Set of Symmetries of the Integrable Equations, Lie Algebra and Non-isospectral Evolution Equations:Ⅱ.AKNS System, J.Phys.A:Math. Gen.19(1986) 3713-3725.
[13]王宝勤,张飞军,源于KdV方程的延拓结构的方程与对称,新疆师范大学学报(自然科学版),第17卷第2期,(1998),1-7.
[14]斯仁道尔吉,组合KdV方程的强对称、对称及其Lie代数,内蒙古师大学报(自然科学汉文版),第3期,数学增刊,(1992),51-57.
[15]陈守婷,半离散AKNS系统的对称与代数结构,博士学位论文,上海大学,(2011).
[16]袁洪芬,超空间上Dirac型方程解的性质,博士学位论文,河北师范大学,(2012).
[17]谷超豪等著,孤立子理论与应用,浙江科学出版社,(1990)216-267.
[18]陈登远编著,孤子引论.-北京：科学出版社,(2006)46-68,187-227.
[19]李翊神编著,孤子与可积系统.上海：上海科技教育出版社,(1999.12)1-40,111-151.
[20]朱国城,关于强对称和遗传对称算子性质的几点注记,科学通报,第2期,(1986).
[21]李翊神,朱国城,可积方程的新的对称、李代数及谱可变演化方程(Ⅰ),中国科学,A辑,第3期,(1987),235-241.
[22]朱国城,李翊神,可积方程的新对称、李代数与谱可变发展方程(Ⅲ),科学通报,第24期,(1986),1845-1849.
[23]马文秀,一类Hamilton算子、遗传对称及可积系,高校应用数学学报,第8卷第1期,(1993),28-35.
[24]王李勇,胡星标,一些非线性演化方程族的新对称,应用数学与计算数学学报,第2卷第2期,(1988),74-82.
[25]Wen-Xiu Ma and Kam-Shun Li, Virasoro Symmetry Algebra of Dirac Soliton Hierarchy, Inverse problems,12(1996)L25-L31.
[26]Jing Yu, Jing-song He, Wen-Xiu Ma, and Yi Cheng, The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac Systems, (2009).
[27]Sirendaoreji, Hamiltonian Structures for the Constrained Flows of the Dirac Hierar-chy, Journal of Inner Monggolia Normal University(Natural Science Edition), Vol.34, No.3(2005).
[28]Yong Fang, Yijun Hou, Huanhe, Jianhai Xue, New Super-hamiltonian Structures of Super-Dirac Hierarchy, Ann.of Diff.Eqs.28:2(2012),164-169.
[29]马文秀,向量场的生成子与发展方程的时间依赖对称,中国科学,第2期,(1991),129-139.
[30]王鸿业,一类系数依赖时间的非线性演化方程的对称及其李代数,郑州大学学报(自然科学版),第27卷第4期,(1995),7-11.
[31]Xin-Zeng Wang and Huan-He Dong, A Lie Superalgebra and Corresponding Hierarchy of Evolution Equations, Modern Physics Letters B, Vol.23, No.28(2009)3387-3396.
[32]Zhu Guocheng, Chen Dengyuan, Zeng Yunbo, A Class Matrix Evolution Equations, Chin.Ann.of Math.3 (1) (1982).
[33]杜殿楼,Lie-Poisson框架下的Dirac-Bargmann系统的可积性,河南科学,第23卷第4期,(2005),472-475.
[34]陶司兴,李超代数与非线性演化方程族的研究,博士学位论文,上海大学,(2011).
[35]连增菊,非线性方程中的对称和对称约化的新进展,硕士学位论文,宁波大学,(2004).