本文選題：孤立子方程 + 零曲率方程��； 參考：《東北師范大學(xué)》2017年博士論文

【摘要】：本論文的主要內(nèi)容分為三部分.第一部分,研究了幾類孤立子可積系及其Hamilton結(jié)構(gòu).首先,在李代數(shù)B2和由它構(gòu)造的李代數(shù)上,選取了兩類滿足屠格式條件的譜矩陣,構(gòu)造了兩類新的具有Hamilton結(jié)構(gòu)的孤立子可積系.其次,考慮李代數(shù)so(4)上的兩組基,得到了兩類不同且均可約化為李代數(shù)so(3)上的孤立子可積系,找到了這兩組基所對(duì)應(yīng)的孤立子可積系之間的關(guān)系.此外,利用李代數(shù)so(4)與李代數(shù)su(2)(?)su(2)同構(gòu),得到了它們對(duì)應(yīng)的孤立子可積系之間的關(guān)系.最后,利用李代數(shù)的Levi分解定理,得到了帶自相容源的廣義AKNS方程族的雙可積耦合和三可積耦合.第二部分,研究了幾類超孤立子可積系及其超Hamilton結(jié)構(gòu).首先,對(duì)于李超代數(shù)spl(2,1),利用超跡恒等式,得到了一類具有超Hamilton結(jié)構(gòu)的(1+1)-維超孤立子可積系,并獲得了譜矩陣的三種達(dá)布變換.利用TAH格式,得到了李超代數(shù)spl(2,1)上的一類(2+1)-維超孤立子可積系及其超Hamilton結(jié)構(gòu).其次,分別給出了李超代數(shù)osp(2,2)和spo(2,2)上的超孤立子可積系及其超雙Hamilton結(jié)構(gòu),且超孤立子可積系均可約化為超AKNS方程族.根據(jù)同構(gòu)關(guān)系spo(2,2)(?)osp(2,2)和spo(2,2)(?)sl(l,2),分別獲得了它們所對(duì)應(yīng)的超孤立子可積系之間的關(guān)系.最后,通過(guò)仿射李超代數(shù)B(0, n)(1)的構(gòu)造方法,給出了李超代數(shù)B(0, n)上的超AKNS方程族及其守恒律.第三部分,研究了雙可積耦合系統(tǒng)的零曲率方程的李代數(shù).首先,討論了雙可積耦合系統(tǒng)的連續(xù)零曲率方程的李代數(shù),并將其應(yīng)用到李代數(shù)so(4)上的廣義等譜與非等譜的孤子族.其次,討論了雙可積耦合系統(tǒng)的離散零曲率方程的李代數(shù),并將其應(yīng)用到廣義等譜與非等譜的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.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.5;O152.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前4條

1 郭�？�;可積的與Hamilton形式的NLS-MKdV方程族[J];數(shù)學(xué)學(xué)報(bào);1997年06期

2 馬文秀;一個(gè)新的Liouville可積的廣義Hamilton方程族及其約化[J];數(shù)學(xué)年刊A輯(中文版);1992年01期

3 曹策問(wèn);AKNS族的Lax方程組的非線性化[J];中國(guó)科學(xué)(A輯 數(shù)學(xué) 物理學(xué) 天文學(xué) 技術(shù)科學(xué));1989年07期

4 曹策問(wèn);保譜方程的換位表示[J];科學(xué)通報(bào);1989年10期

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本文編號(hào)：1748972