李倩,夏鐵成

(1.上海大學理學院,上海 200444;

2.鄭州航空工業(yè)管理學院理學院,鄭州 450005)

Dirac孤子族的三可積耦合及其雙Hamiltonian結構

李倩1,2,夏鐵成1

(1.上海大學理學院,上海 200444;

2.鄭州航空工業(yè)管理學院理學院,鄭州 450005)

基于擴大的零曲率方程和矩陣李代數的半直和,得到了Dirac孤子族的三可積耦合,并借助變分恒等式得到了三可積耦合的雙Hamiltonian結構.

Dirac孤子族;三可積耦合;雙Hamiltonian結構

眾所周知,可積耦合是孤子理論中有趣而重要的課題[1-3].研究可積耦合不僅可以概括對稱問題,而且為可積系統(tǒng)的完全分類提供了線索,甚至可以顯現出可積方程所擁有的數學結構.自從可積耦合的定義[4]被提出后,方程族可積耦合得到廣泛關注,并出現了很多建立方程組可積耦合的方法,如擴大譜問題、李代數的半直和、新的李代數等[5-20].關于可積耦合的研究甚至推廣到了超可積系統(tǒng)[18-20].2012年,Ma[21]用塊型矩陣李代數建立了方程族的雙可積耦合,之后又進一步將雙可積耦合推廣到三可積耦合[22].

對于給定的可積系統(tǒng)

式中,u為獨立變量構成的列向量.可積系統(tǒng)(1)的三可積耦合指的是擴大的三角形可積系統(tǒng):

如果S1(u,u1),S2(u,u1,u2)和S3(u,u1,u2,u3)中至少有一個關于任何新的獨立變量u1,u2,u3是非線性的,則稱這個系統(tǒng)是非線性可積耦合的.

為了建立三可積耦合,需要一系列三角形4×4的分塊矩陣M(A1,A2,A3,A4),其中Ai(1≤i≤4)是同階的方陣.引入一系列具有半直和分解的矩陣李代數：

其中Ai(λ)(i=1,2,3,4)為λ的羅朗級數,則其必為非半單的.顯然,gc是的一個非平凡理想.

上述所提出的李代數為產生非線性三可積耦合提供了一組基,因為交換算子[A2,B2] 和[A3,B3]在對應的三可積耦合中能夠產生非線性項,而其余現有的李代數產生線性可積耦合,其中子塊A1對應原始的可積系統(tǒng),子塊A2,A3和A4用來生成輔助向量域S1,S2和S3.

本工作的主要目的是基于文獻[23]中的方法建立Dirac族的三可積耦合.選取其中一個分塊矩陣:

式中,α,β和μ是3個任意給定的常數.進一步地,希望獲得對應三可積耦合的雙Hamiltonian結構.

1 Dirac族

2 Dirac族的三可積耦合

基于特殊的非半單李代數,選取如下擴大的譜矩陣:

其中si,vi是新的獨立變量.為了求解擴大的零曲率方程

當m≥2時,由式(31)可以得出Dirac方程族的非線性三可積耦合.

3 Hamiltonian結構

應用變分恒等式(23)建立對應三可積耦合的Hamiltonian結構:

要求FT=F.在對稱條件下,不變性質〈a,[b,c]〉=〈[a,b],c〉等價于要求F(R(b))T=?R(b)F, b∈R12.具有任意常數b的矩陣方程產生矩陣F的線性系統(tǒng),求解對應的系統(tǒng)可以得到

[1]ABLOWITZ M,LADIK J F.Nonlinear differential-difference equation[J].Journal of Mathematical Physics,1975,16(3):598-603.

[2]TU G Z.A trace identity and its applications to the theory of discrete integrable systems[J]. Journal of Physics A:Mathematical and General,1990,23(17):3903-3922.

[3]OHTA Y,HIROTA R.A discrete KdV equation and its Casorati determinant solution[J].Journal of the Physical Society of Japan,1991,60(6):2095.

[4]MA W X,FUCHSSTEINER B.Integrable theory of the perturbation equations[J].Chaos, Solitons&Fractals,1996,7(8):1227-1250.

[5]MA W X,XU X X,ZHANG Y F.Semi-direct sums of Lie algebras and continuous integrable couplings[J].Physics Letter A,2006,351(3):125-130.

[6]MA W X.Enlarging spectral problems to construct integrable couplings of soliton equations[J]. Physics Letters A,2003,316(2):72-76.

[7]ZHANG Y F.A generalized multi-component Glachette-Johnson(GJ)hierarchy and its integrable coupling system[J].Chaos,Solitons&Fractals,2004,21(2):305-310.

[8]GUO F K,ZHANG Y F.A new loop algebra and a corresponding integrable hierarchy,as well as its integrable coupling[J].Journal of Mathematical Physics,2003,44(12):5793-5803.

[9]XIA T C,YOU F C.Generalized multi-component TC hierarchy and its multi-component integrable coupling system[J].Communications in Theoretical Physics,2005,44(5):793-798.

[10]MA W X.Nonlinear continuous integrable Hamiltonian couplings[J].Applied Mathematics and Computation,2011,217(17):7238-7244.

[11]ZHANG Y F,TAM H.Four Lie algebras associated with R6and their applications[J].Journal of Mathematical Physics,2010,DOI:10.1063/1.3489126.

[12]MA W X,ZHU Z N.Constructing nonlinear discrete integrable Hamiltonian couplings[J].Computers&Mathematics with Applications,2010,60(19):2601-2608.

[13]YU F J.A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure[J].Physics Letters A,2011,375(13):1504-1509.

[14]ZHANG Y F.Lie algebras for constructing nonlinear integrable couplings[J].Communications in Theoretical Physics,2011,56(5):805-812.

[15]ZHANG Y F,FAN E G.Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti-Pempinelli-Tu hierarchy[J].Journal of Mathematical Physics,2010,DOI: 10.1063/1.3462736.

[16]GUO F K,ZHANG Y F.The quadratic-form identity for constructing the Hamiltonian structure of integrable systems[J].Journal of Physics A:Mathematical and General,2005,38:8537-8548.

[17]ZHANG Y F,ZHANG H Q.A direct method for integrable couplings of TD hierarchy[J].Journal of Mathematical Physics,2002,43(1):466-472.

[18]YOU F C.Nonlinear super integrable couplings of super Dirac hierarchy and its super Hamiltonian structures[J].Communications in Theoretical Physics,2012,57(6):961-966.

[19]MA W X,HE J S,QIN Z Y.A super trace identity and its applications to super integrable systems[J].Journal of Mathematical Physics,2008,DOI:10.1063/1.2897036.

[20]YOU F C.Nonlinear super integrable Hamiltonian couplings[J].Journal of Mathematical Physics,2011,DOI:10.1063/1.3669484.

[21]MA W X.Loop algebras and bi-integrable couplings[J].Chinese Annals of Mathematics B, 2012,33(2):207-224.

[22]MENG J,MA W X.Hamiltonian tri-integrable couplings of the AKNS hierarchy[J].Communications in Theoretical Physics,2013,59(4):385-392.

[23]MA W X.Variational identities and applications to Hamiltonian structures of soliton equations[J].Nonlinear Analysis,2009,71(12):1716-1726.

[24]OLVER P J.Applications of Lie groups to differential equations[M].New York:Springer-Verlag, 1986.

Tri-integrable couplings of Dirac hierarchy and its bi-Hamiltonian structure

LI Qian1,2,XIA Tiecheng1
(1.College of Sciences,Shanghai University,Shanghai 200444,China; 2.College of Sciences,Zhengzhou University of Aeronautics,Zhengzhou 450005,China)

Tri-integrable couplings of Dirac hierarchy are obtained based on the enlarged zero curvature equation from semi-direct sums of Lie algebras.Its bi-Hamiltonian structures are then established with variational identity.

Dirac hierarchy;tri-integrable couplings;bi-Hamiltonian structure

O 175.2

A

1007-2861(2017)02-0257-10

10.3969/j.issn.1007-2861.2015.04.022

2015-10-26

國家自然科學基金資助項目(11271008,61640315);河南省高等學校重點科研資助項目(17A120006)

夏鐵成(1960—),男,教授,博士生導師,研究方向為孤子與可積系統(tǒng).E-mail:xiatc@shu.edu.cn