一类广义三阶演化方程的半单李代数分类
发布时间：2018-11-11 01:35
【摘要】：本文主要研究了一类一维空间变量的广义三阶演化方程ut=F(t,x,u,ux,uxx,,uxxx)的半单李代数的对称群分类问题,分类过程包括以下三步:首先,构造方程的等价群、其所容许的一般形式的对称群和关于未知函数F的分类方程;其次,结合抽象李代数的结构,用已构造的一般形式的李对称生成算子来实现方程可能容许的所有李对称群;最后,将已得对称群的生成算子代入分类方程并求解所得偏微分方程组,可得关于未知函数F的表达式,解之可得相应的不变方程.通过对称群分类方法,我们构造了容许半单李代数和可解李代数的半直和的所有不等价方程.这些方程所满足的对称性质可归纳如下:●得到了两个容许一维李代数的不变方程;·给出了一个容许同构于A2,2的二维可解李代数的不变方程;·得到了了六个容许同构于sl(2,R)的半单李代数的不变方程;●给出了十九个容许包含非平凡Levi因子的李代数的不变方程.本文的各章节内容安排如下:第一章:简要地介绍了群分类问题的方法和意义,以及目前的研究进展和已有研究成果;第二章:给出了有关一类广义三阶演化方程的半单李代数分类所需的相关定义、定理及推论;第三章:得到了容许半单李代数的一类广义三阶演化方程的所有不等价方程;第四章:在对本文研究工作总结的基础上,提出了今后的研究内容。
[Abstract]:In this paper, we study the problem of symmetric group classification of semi-simple lie algebras for a class of generalized third-order evolution equations ut=F (TX). The classification process consists of the following three steps: firstly, the equivalent groups of the equations are constructed. The general form of symmetric group and the classification equation about the unknown function F. Secondly, combining the structure of abstract lie algebra, we use the constructed general form of lie symmetric generating operator to realize all the lie symmetric groups that the equation may allow. Finally, the generating operator of the obtained symmetric group is substituted into the classification equation and the system of partial differential equations is solved. The expression of the unknown function F can be obtained, and the corresponding invariant equation can be obtained. By means of the classification method of symmetric groups, we construct all the inequivalent equations of the semidirect sum of admissible semisimple lie algebras and solvable lie algebras. The symmetry properties of these equations can be summarized as follows: two invariant equations of admissible one-dimensional lie algebras are obtained, and an invariant equation of two-dimensional solvable lie algebras with admissible isomorphism is given. In this paper, we obtain six invariant equations of semisimple lie algebras with admissible isomorphism to sl (2R), and give the invariant equations of 19 admissible lie algebras containing nontrivial Levi factors. The main contents of this paper are as follows: chapter 1: briefly introduces the methods and significance of group classification, as well as the current research progress and existing research results; In chapter 2, we give the necessary definitions, theorems and corollaries for the classification of semi-simple lie algebras for a class of generalized third-order evolution equations, and obtain all the inequivalent equations of a class of generalized third-order evolution equations with admissible semi-simple lie algebras. Chapter four: on the basis of summarizing the research work of this paper, the future research contents are put forward.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.5
本文编号：2324130
[Abstract]:In this paper, we study the problem of symmetric group classification of semi-simple lie algebras for a class of generalized third-order evolution equations ut=F (TX). The classification process consists of the following three steps: firstly, the equivalent groups of the equations are constructed. The general form of symmetric group and the classification equation about the unknown function F. Secondly, combining the structure of abstract lie algebra, we use the constructed general form of lie symmetric generating operator to realize all the lie symmetric groups that the equation may allow. Finally, the generating operator of the obtained symmetric group is substituted into the classification equation and the system of partial differential equations is solved. The expression of the unknown function F can be obtained, and the corresponding invariant equation can be obtained. By means of the classification method of symmetric groups, we construct all the inequivalent equations of the semidirect sum of admissible semisimple lie algebras and solvable lie algebras. The symmetry properties of these equations can be summarized as follows: two invariant equations of admissible one-dimensional lie algebras are obtained, and an invariant equation of two-dimensional solvable lie algebras with admissible isomorphism is given. In this paper, we obtain six invariant equations of semisimple lie algebras with admissible isomorphism to sl (2R), and give the invariant equations of 19 admissible lie algebras containing nontrivial Levi factors. The main contents of this paper are as follows: chapter 1: briefly introduces the methods and significance of group classification, as well as the current research progress and existing research results; In chapter 2, we give the necessary definitions, theorems and corollaries for the classification of semi-simple lie algebras for a class of generalized third-order evolution equations, and obtain all the inequivalent equations of a class of generalized third-order evolution equations with admissible semi-simple lie algebras. Chapter four: on the basis of summarizing the research work of this paper, the future research contents are put forward.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.5
【参考文献】
相关期刊论文 前1条
1 ;Lie group classification of the N-th-order nonlinear evolution equations[J];Science China(Mathematics);2011年12期
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