一個(gè)新的Lie代數(shù)和它的應(yīng)用

常雙領(lǐng)

(北京物資學(xué)院信息學(xué)院,北京 101149)

一個(gè)新的Lie代數(shù)和它的應(yīng)用

常雙領(lǐng)

(北京物資學(xué)院信息學(xué)院,北京 101149)

通過構(gòu)造一個(gè)新的Lie代數(shù),利用它相應(yīng)的Loop代數(shù)設(shè)計(jì)等譜Lax對(duì),根據(jù)其相容性條件,得到了一族Lax可積方程族,其一種約化形式為著名的AKNS族.根據(jù)跡恒等式得到該方程族的Hamilton結(jié)構(gòu).利用該可積方程族可以進(jìn)一步研究它的達(dá)布變換、對(duì)稱、代數(shù)幾何解等相關(guān)性質(zhì).

Lie代數(shù);可積方程族;跡恒等式;Hamilton結(jié)構(gòu)

1 引言

借助于Lie代數(shù)設(shè)計(jì)等譜問題,根據(jù)相容性條件可以獲得一系列的Lax可積的方程族.一些科研工作者經(jīng)過辛勤的工作已經(jīng)獲得一些可積Hamilton方程族[1-8],并且研究了他們的非線性化、守恒律、達(dá)布變換等[9-10].尋求新的高維Lie代數(shù)和具有物理意義的可積系統(tǒng)是孤立子理論研究中的基本工作之一.本文首先構(gòu)造了一個(gè)新的Lie代數(shù),設(shè)計(jì)了一個(gè)含有六個(gè)位勢(shì)函數(shù)的等譜問題,根據(jù)相容性條件和跡恒等式,獲得具有Hamilton結(jié)構(gòu)的可積方程族.該方程族是著名的AKNS方程族的擴(kuò)展,是對(duì)可積系統(tǒng)的進(jìn)一步完善和發(fā)展.

2 一個(gè)八維Lie代數(shù)和它的分解子Lie代數(shù)

3 一個(gè)新的可積的方程族和它的約化

4 方程族(4)的Hamilton結(jié)構(gòu)

參考文獻(xiàn)

[1]Guizhang Tu.The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J.Math.Phys.,1989,302:330-338.

[2]郭福奎.Loop代數(shù)~A1的子代數(shù)與可積Hamilton方程族[J].數(shù)學(xué)物理學(xué)報(bào),1999,19(5):507-512.

[3]Guo Fukui,Zhang Yufeng.Two unifed formulae[J].Physics Letters A,2007,366:403-410.

[4]Zeng Yunbo,Xiu Wen,Lin Runliang.Integration of the soliton hierarchy with self-consistent Sources[J]. Journal of Mathematical Physics,2000,408:5453-5490.

[5]Zhang Yufeng,Tam Honwah.A few integrable systems and spatial spectral transformations[J].Commun. Nonlinear Sci.Numer.Simulat.,2009,14:3770-3783.

[6]Chang Hui,Zhang Yufeng.A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure[J].Chaos,Solitons and Fractals, 2009,39:473-478.

[7]梅建琴.4×4B-型KdV方程族[J].數(shù)學(xué)年刊:A輯,2012,33(2):161-174.

[8]夏鐵成,張登朋,特木爾朝魯.帶有自相容源的屠族新可積耦合[J].數(shù)學(xué)物理學(xué)報(bào):A輯,2012,32(5):941-949.

[9]Zhang Yu,Chang Hui,Li Na.Explicit N-fold Darboux transformation for the classical Boussinesq system and multi-soliton solutions[J].Physics Letters A,2009,373:454-457.

[10]Qu Changzheng,Yao Ruoxia.Liu Ruochen.Multi-component WKI equations and their conservation laws[J].Physics Letters A,2004,331:325-331.

A new Lie algebra and its applications

Chang Shuangling

(School of Information,Beijing Wuzi University,Beijing101149,China)

By constructing a new Lie algebra and its corresponding Loop algebra,an isospectral Lax pair is established whose compatibility condition gives rise to a Lax integrable hierarcy,whose reduced form is the well-known AKNS hierarchy.Its Hamilton structure is obtained by the use of the trace identity.Then,its Darboux transformations,symmetry,algebro-geometric solutions,and so on will be investigated further.

Lie algebra,trace identity,integrable hierarchy,Hamilton structure

O152

A

1008-5513(2013)06-0627-07

10.3969/j.issn.1008-5513.2013.06.012

2013-09-05.

常雙領(lǐng)(1975-),碩士,研究方向:基礎(chǔ)數(shù)學(xué).

2010 MSC:45G15