李（超）代数在可积系及其Hamilton结构中的应用

发布时间：2018-04-14 10:36

本文选题：孤立子方程 + 零曲率方程　；参考：《东北师范大学》2017年博士论文

【摘要】：本论文的主要内容分为三部分.第一部分,研究了几类孤立子可积系及其Hamilton结构.首先,在李代数B2和由它构造的李代数上,选取了两类满足屠格式条件的谱矩阵,构造了两类新的具有Hamilton结构的孤立子可积系.其次,考虑李代数so(4)上的两组基,得到了两类不同且均可约化为李代数so(3)上的孤立子可积系,找到了这两组基所对应的孤立子可积系之间的关系.此外,利用李代数so(4)与李代数su(2)(?)su(2)同构,得到了它们对应的孤立子可积系之间的关系.最后,利用李代数的Levi分解定理,得到了带自相容源的广义AKNS方程族的双可积耦合和三可积耦合.第二部分,研究了几类超孤立子可积系及其超Hamilton结构.首先,对于李超代数spl(2,1),利用超迹恒等式,得到了一类具有超Hamilton结构的(1+1)-维超孤立子可积系,并获得了谱矩阵的三种达布变换.利用TAH格式,得到了李超代数spl(2,1)上的一类(2+1)-维超孤立子可积系及其超Hamilton结构.其次,分别给出了李超代数osp(2,2)和spo(2,2)上的超孤立子可积系及其超双Hamilton结构,且超孤立子可积系均可约化为超AKNS方程族.根据同构关系spo(2,2)(?)osp(2,2)和spo(2,2)(?)sl(l,2),分别获得了它们所对应的超孤立子可积系之间的关系.最后,通过仿射李超代数B(0, n)(1)的构造方法,给出了李超代数B(0, n)上的超AKNS方程族及其守恒律.第三部分,研究了双可积耦合系统的零曲率方程的李代数.首先,讨论了双可积耦合系统的连续零曲率方程的李代数,并将其应用到李代数so(4)上的广义等谱与非等谱的孤子族.其次,讨论了双可积耦合系统的离散零曲率方程的李代数,并将其应用到广义等谱与非等谱的Toda孤子族.
[Abstract]:The main content of this paper is divided into three parts.In the first part, we study several kinds of soliton integrable systems and their Hamilton structures.Firstly, on the lie algebra B2 and the lie algebra constructed by it, two classes of spectral matrices satisfying the conditions of Tu format are selected, and two new classes of soliton integrable systems with Hamilton structure are constructed.Secondly, considering two groups of bases on lie algebra SO4), we obtain two classes of soliton integrable systems which can be reduced to lie algebras so _ 3), and find the relation between the soliton integrable systems corresponding to these two groups of bases.In addition, by using the isomorphism of lie algebra so _ 4) and lie algebra Su ~ (2), we obtain the relation between their corresponding soliton integrable systems.Finally, by using the Levi decomposition theorem of lie algebra, the double integrable coupling and three integrable coupling of the family of generalized AKNS equations with self-compatible sources are obtained.In the second part, several kinds of supersoliton integrable systems and their super Hamilton structures are studied.First of all, for Li Chao algebraic splitt2n, we obtain a class of hypersoliton integrable systems with super Hamilton structure by using hypertrace identities, and obtain three kinds of Darboux transformations of spectral matrices.By using the TAH scheme, we obtain a class of hypersoliton integrable systems and their super Hamilton structures on Li Chao algebras spl2 + 1).Secondly, we give the supersoliton integrable system and its super-double Hamilton structure on Li Chao algebras ospt2 ~ (2) and spoo _ 2 ~ (2), respectively, and the supersoliton integrable system can be reduced to a family of super AKNS equations.According to the isomorphism relationship, the relations between the supersoliton integrable systems are obtained, respectively.Finally, by means of the construction method of affine Li Chao algebras BX 0, NX 1), the super equation family and its conservation laws are given on Li Chao algebra BX 0, n).In the third part, we study the lie algebra of the zero curvature equation of the double integrable coupled system.First, we discuss the lie algebra of the continuous zero curvature equation for a biintegrable coupled system, and apply it to the generalized isospectral and nonisospectral soliton families on the lie algebra so4).Secondly, we discuss the lie algebra of the discrete zero curvature equation for the double integrable coupled system, and apply it to the generalized isospectral and nonisospectral Toda soliton families.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.5;O152.5

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