一個新的Lie代數和它的應用

常雙領

(北京物資學院信息學院,北京 101149)

一個新的Lie代數和它的應用

常雙領

(北京物資學院信息學院,北京 101149)

通過構造一個新的Lie代數,利用它相應的Loop代數設計等譜Lax對,根據其相容性條件,得到了一族Lax可積方程族,其一種約化形式為著名的AKNS族.根據跡恒等式得到該方程族的Hamilton結構.利用該可積方程族可以進一步研究它的達布變換、對稱、代數幾何解等相關性質.

Lie代數;可積方程族;跡恒等式;Hamilton結構

1 引言

借助于Lie代數設計等譜問題,根據相容性條件可以獲得一系列的Lax可積的方程族.一些科研工作者經過辛勤的工作已經獲得一些可積Hamilton方程族[1-8],并且研究了他們的非線性化、守恒律、達布變換等[9-10].尋求新的高維Lie代數和具有物理意義的可積系統是孤立子理論研究中的基本工作之一.本文首先構造了一個新的Lie代數,設計了一個含有六個位勢函數的等譜問題,根據相容性條件和跡恒等式,獲得具有Hamilton結構的可積方程族.該方程族是著名的AKNS方程族的擴展,是對可積系統的進一步完善和發展.

2 一個八維Lie代數和它的分解子Lie代數

3 一個新的可積的方程族和它的約化

4 方程族(4)的Hamilton結構

參考文獻

[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~A1的子代數與可積Hamilton方程族[J].數學物理學報,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].數學年刊: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].數學物理學報: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(1975-),碩士,研究方向:基礎數學.

2010 MSC:45G15