李倩,夏鐵成

(1.上海大學(xué)理學(xué)院,上海 200444;

2.鄭州航空工業(yè)管理學(xué)院理學(xué)院,鄭州 450005)

Dirac孤子族的三可積耦合及其雙Hamiltonian結(jié)構(gòu)

李倩1,2,夏鐵成1

(1.上海大學(xué)理學(xué)院,上海 200444;

2.鄭州航空工業(yè)管理學(xué)院理學(xué)院,鄭州 450005)

基于擴(kuò)大的零曲率方程和矩陣?yán)畲鷶?shù)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(jié)構(gòu).

Dirac孤子族;三可積耦合;雙Hamiltonian結(jié)構(gòu)

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(duì)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(xiàn)出可積方程所擁有的數(shù)學(xué)結(jié)構(gòu).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(guān)注,并出現(xiàn)了很多建立方程組可積耦合的方法,如擴(kuò)大譜問題、李代數(shù)的半直和、新的李代數(shù)等[5-20].關(guān)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?yán)畲鷶?shù)建立了方程族的雙可積耦合,之后又進(jìn)一步將雙可積耦合推廣到三可積耦合[22].

對(duì)于給定的可積系統(tǒng)

式中,u為獨(dú)立變量構(gòu)成的列向量.可積系統(tǒng)(1)的三可積耦合指的是擴(kuò)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(gè)關(guān)于任何新的獨(dú)立變量u1,u2,u3是非線性的,則稱這個(gè)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?yán)畲鷶?shù)：

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

上述所提出的李代數(shù)為產(chǎn)生非線性三可積耦合提供了一組基,因?yàn)榻粨Q算子[A2,B2] 和[A3,B3]在對(duì)應(yīng)的三可積耦合中能夠產(chǎn)生非線性項(xiàng),而其余現(xiàn)有的李代數(shù)產(chǎn)生線性可積耦合,其中子塊A1對(duì)應(yīng)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(xiàn)[23]中的方法建立Dirac族的三可積耦合.選取其中一個(gè)分塊矩陣:

式中,α,β和μ是3個(gè)任意給定的常數(shù).進(jìn)一步地,希望獲得對(duì)應(yīng)三可積耦合的雙Hamiltonian結(jié)構(gòu).

1 Dirac族

2 Dirac族的三可積耦合

基于特殊的非半單李代數(shù),選取如下擴(kuò)大的譜矩陣:

其中si,vi是新的獨(dú)立變量.為了求解擴(kuò)大的零曲率方程

當(dāng)m≥2時(shí),由式(31)可以得出Dirac方程族的非線性三可積耦合.

3 Hamiltonian結(jié)構(gòu)

應(yīng)用變分恒等式(23)建立對(duì)應(yīng)三可積耦合的Hamiltonian結(jié)構(gòu):

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

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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(guó)家自然科學(xué)基金資助項(xiàng)目(11271008,61640315);河南省高等學(xué)校重點(diǎn)科研資助項(xiàng)目(17A120006)

夏鐵成(1960—),男,教授,博士生導(dǎo)師,研究方向?yàn)楣伦优c可積系統(tǒng).E-mail:xiatc@shu.edu.cn