一類廣義三階演化方程的半單李代數(shù)分類
發(fā)布時(shí)間：2018-11-11 01:34
【摘要】：本文主要研究了一類一維空間變量的廣義三階演化方程ut=F(t,x,u,ux,uxx,,uxxx)的半單李代數(shù)的對稱群分類問題,分類過程包括以下三步:首先,構(gòu)造方程的等價(jià)群、其所容許的一般形式的對稱群和關(guān)于未知函數(shù)F的分類方程;其次,結(jié)合抽象李代數(shù)的結(jié)構(gòu),用已構(gòu)造的一般形式的李對稱生成算子來實(shí)現(xiàn)方程可能容許的所有李對稱群;最后,將已得對稱群的生成算子代入分類方程并求解所得偏微分方程組,可得關(guān)于未知函數(shù)F的表達(dá)式,解之可得相應(yīng)的不變方程.通過對稱群分類方法,我們構(gòu)造了容許半單李代數(shù)和可解李代數(shù)的半直和的所有不等價(jià)方程.這些方程所滿足的對稱性質(zhì)可歸納如下:●得到了兩個(gè)容許一維李代數(shù)的不變方程;·給出了一個(gè)容許同構(gòu)于A2,2的二維可解李代數(shù)的不變方程;·得到了了六個(gè)容許同構(gòu)于sl(2,R)的半單李代數(shù)的不變方程;●給出了十九個(gè)容許包含非平凡Levi因子的李代數(shù)的不變方程.本文的各章節(jié)內(nèi)容安排如下:第一章:簡要地介紹了群分類問題的方法和意義,以及目前的研究進(jìn)展和已有研究成果;第二章:給出了有關(guān)一類廣義三階演化方程的半單李代數(shù)分類所需的相關(guān)定義、定理及推論;第三章:得到了容許半單李代數(shù)的一類廣義三階演化方程的所有不等價(jià)方程;第四章:在對本文研究工作總結(jié)的基礎(chǔ)上,提出了今后的研究內(nèi)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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O152.5
本文編號(hào)：2324131
[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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O152.5
【參考文獻(xiàn)】
相關(guān)期刊論文 前1條
1 ;Lie group classification of the N-th-order nonlinear evolution equations[J];Science China(Mathematics);2011年12期
，本文編號(hào)：2324131
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