孤子方程族的可积耦合系统和分数阶Hamiltonian结构
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摘要
本文研究的主要内容包括：运用李代数，首先给出一些方程族的可积耦合系统的构造模式，并且给出了非等谱情形的离散可积耦合系统。进而讨论了连续和离散方程族的零曲率表示的李代数结构。另外，还介绍了孤子族的生成及Hamiltonian结构，Liouville可积性。最后利用分数阶微积分给出了孤子方程的分数阶Hamiltonian结构。其具体内容为：
第一章介绍了孤立子理论，可积系统，非线性发展方程精确求解，分数阶微积分的历史发展及研究现状，同时介绍了国内外学者在这方面取得的成果。
第二章简要的介绍了Kac-Moody代数，Hamiltonian函数的概念及相关的性质。详细的阐述和介绍了AC=BD理论中的一些相关的定理和性质及其在这个框架下的一些重要应用。
第三章首先从新的谱问题出发导出一族矩阵Lax可积方程族，并获得它的Hamiltonian结构。另外从Lax对出发，采用提出的谱扩张方法得到了许多新的可积耦合方程族，在此基础上，把这种方法推广到高维空间，并获得了一系列的多分量可积耦合方程族。但是利用这种方法不能得到可积耦合方程族的Hamiltonian结构(尤其是多分量可积耦合方程族的Hamiltonian结构)，针对此问题，文中给出广义的killing内积，并且运用广义的二次迹恒等式获得了多分量耦合系统的Hamiltonian结构。其中给出了多分量Jaulent-Miodek方程族，多分量2+1维GJ方程族和耦合Dirac方程族的Hamiltonian结构。另外利用一个广义的矩阵谱问题，得到了耦合方程族的R-矩阵。其中以AKNS族为例，得到了耦合AKNS方程族的R-矩阵。
第四章从loop代数(?)_1的一个子代数出发，利用屠格式求出了一类离散情形Lax可积耦合的系统，并且得到非等谱的离散可积方程族和耦合系统，另外我们还提出了2+1维非等谱离散可积耦合形式，利用谱参数λ满足的非等谱条件，得到了Blaszak-Marciniak晶格方程的耦合系统。国际著名杂志《Physics Letters A》的编委A.R.Bishop对此种方法给出了很好的评价“The method gives two kinds of classification to a soliton equation，itis an interesting and important work”。另外，进一步考虑了离散系统Darboux变换。最后讨论了离散可积方程与连续可积方程的联系，通常人们采用的是对势函数作变换，而文中采用对算子作变换，利用计算机软件通过比较算子的系数，得到了很好的结果，并且把一个新的离散方程转化成AKNS方程。这样做不仅可以建立离散与连续方程之间的关系，更重要的是可以通过连续型方程的精确解(解析解)获得相应的离散方程的数值解，这样就可以得到更多，更好的数值解。
第五章在整数情形可积系统的基础上，进一步考虑分数形的Hamiltonian结构，文中运用了外微分与分数阶微积分结合，给出了分数空间和分数形式的Hamiltonian形式。在这里主要考虑要把整数情形的结论发展到分数情形，建立一套分数阶Hamiltonian结构和可积系统。我们已经完成了分数阶零曲率方程的构造，得到了分数阶情形的AKNS方程和C-KdV方程，并且给出了它们简单形式的Hamiltonian函数。另外利用Riemann-Liouville分数阶算子和分数形式的Possion括号，把Hamiltonian结构的辛形式推广到分数阶情形。
The major contents in this dissertation include: the generation of soliton hierarchies ofequation and the structure equations of Lie algebra, Hamiltonian structures, Liouville integra-bility and integrable coupling system, some isospectral and nonisospectral siliton equations arestudied in 2+1 dimensions. Finally, the fractional Hamiltonian structures of soliton equationare worked out by using of the nonlinear fractional differential operators.
Chapter 1 is devoted to reviewing the history and development of the soliton theory, inte-grable system, solving nonlinear evolution equations and fractional calculous, with an emphasison some achievements on the subjects involved in this dissertation are presented at home andabroad.
Chapter 2 introduces Kac-Moody algebra, some basic notations and properties on Hamil-tonian function, basic theories on AC=BD as well as the construction of exact solutions ofnonlinear evolution equation(s) under the instruction of this theory.
Chapter 3 first presents a matrix Lax integrable equation hierarchy by using of new matrixspectral problem, then obtains its Hamiltonian structure. According to the Lax pair, somenew integrable coupling equation hierarchies are worked out by using of the enlarged spectralproblem. This way is used to high dimensions space and obtain a few of multi-componentintegrable coupling equation hierarchies. The generalized killing inner product is presented,the Hamiltonian structure of multi-component integrable coupling system is solved by using ofgeneralized quadratic-form identity. The Hamiltonian structures of multi-component Jaulent-Miodeck equation hierarchy, multi-component 2+1 dimensional GJ equation hierarchy and thecoupling Dirac equation hierarchy are considered. The R-matrix of coupling equation hierarchy isobtained through a generalized matrix spectral problem. For example, the R-matrix of couplingAKNS equation hierarchy is given.
Chapter 4 investigates the discrete nonisospectral problem. First, basing on loop algebraA1, a new sub-algebra is presented. The isospectral and nonisospectral Lax integrable couplingequation hierarchies are worked out. Second, the 2+1-dimension nonisospectral integrable cou-pling model is presented, the nonisospectral integrable coupling system of Blaszak-Marciniaklattice hierarchy is obtained under the nonisospectral condition of spectral parameterλ. TheA.R. Bishop Editor-in-Chief of《Physics Letters A》appraised that " The method gives twokinds of classification to a soliton equation, it is an interesting and important work ". Darbouxtransformation is considered. Last, the relation between the discrete soliton equation and the AKNS soliton equation hierarchy is given through the transform of potential functions.
Chapter 5, the fractional Hamiltonian structure of soliton equation is considered. Thefractional zero curvature equation is constructed by using of fractional differential formula, andobtain the fractional AKNS and fractional C-KdV equation by making use of the fractional zerocurvature equation, then their Hamiltonian structures are worked out. The fractional Poissonbracket is defined, a Hamiltonian system of fractional form is presented.
第一章介绍了孤立子理论，可积系统，非线性发展方程精确求解，分数阶微积分的历史发展及研究现状，同时介绍了国内外学者在这方面取得的成果。
第二章简要的介绍了Kac-Moody代数，Hamiltonian函数的概念及相关的性质。详细的阐述和介绍了AC=BD理论中的一些相关的定理和性质及其在这个框架下的一些重要应用。
第三章首先从新的谱问题出发导出一族矩阵Lax可积方程族，并获得它的Hamiltonian结构。另外从Lax对出发，采用提出的谱扩张方法得到了许多新的可积耦合方程族，在此基础上，把这种方法推广到高维空间，并获得了一系列的多分量可积耦合方程族。但是利用这种方法不能得到可积耦合方程族的Hamiltonian结构(尤其是多分量可积耦合方程族的Hamiltonian结构)，针对此问题，文中给出广义的killing内积，并且运用广义的二次迹恒等式获得了多分量耦合系统的Hamiltonian结构。其中给出了多分量Jaulent-Miodek方程族，多分量2+1维GJ方程族和耦合Dirac方程族的Hamiltonian结构。另外利用一个广义的矩阵谱问题，得到了耦合方程族的R-矩阵。其中以AKNS族为例，得到了耦合AKNS方程族的R-矩阵。
第四章从loop代数(?)_1的一个子代数出发，利用屠格式求出了一类离散情形Lax可积耦合的系统，并且得到非等谱的离散可积方程族和耦合系统，另外我们还提出了2+1维非等谱离散可积耦合形式，利用谱参数λ满足的非等谱条件，得到了Blaszak-Marciniak晶格方程的耦合系统。国际著名杂志《Physics Letters A》的编委A.R.Bishop对此种方法给出了很好的评价“The method gives two kinds of classification to a soliton equation，itis an interesting and important work”。另外，进一步考虑了离散系统Darboux变换。最后讨论了离散可积方程与连续可积方程的联系，通常人们采用的是对势函数作变换，而文中采用对算子作变换，利用计算机软件通过比较算子的系数，得到了很好的结果，并且把一个新的离散方程转化成AKNS方程。这样做不仅可以建立离散与连续方程之间的关系，更重要的是可以通过连续型方程的精确解(解析解)获得相应的离散方程的数值解，这样就可以得到更多，更好的数值解。
第五章在整数情形可积系统的基础上，进一步考虑分数形的Hamiltonian结构，文中运用了外微分与分数阶微积分结合，给出了分数空间和分数形式的Hamiltonian形式。在这里主要考虑要把整数情形的结论发展到分数情形，建立一套分数阶Hamiltonian结构和可积系统。我们已经完成了分数阶零曲率方程的构造，得到了分数阶情形的AKNS方程和C-KdV方程，并且给出了它们简单形式的Hamiltonian函数。另外利用Riemann-Liouville分数阶算子和分数形式的Possion括号，把Hamiltonian结构的辛形式推广到分数阶情形。
The major contents in this dissertation include: the generation of soliton hierarchies ofequation and the structure equations of Lie algebra, Hamiltonian structures, Liouville integra-bility and integrable coupling system, some isospectral and nonisospectral siliton equations arestudied in 2+1 dimensions. Finally, the fractional Hamiltonian structures of soliton equationare worked out by using of the nonlinear fractional differential operators.
Chapter 1 is devoted to reviewing the history and development of the soliton theory, inte-grable system, solving nonlinear evolution equations and fractional calculous, with an emphasison some achievements on the subjects involved in this dissertation are presented at home andabroad.
Chapter 2 introduces Kac-Moody algebra, some basic notations and properties on Hamil-tonian function, basic theories on AC=BD as well as the construction of exact solutions ofnonlinear evolution equation(s) under the instruction of this theory.
Chapter 3 first presents a matrix Lax integrable equation hierarchy by using of new matrixspectral problem, then obtains its Hamiltonian structure. According to the Lax pair, somenew integrable coupling equation hierarchies are worked out by using of the enlarged spectralproblem. This way is used to high dimensions space and obtain a few of multi-componentintegrable coupling equation hierarchies. The generalized killing inner product is presented,the Hamiltonian structure of multi-component integrable coupling system is solved by using ofgeneralized quadratic-form identity. The Hamiltonian structures of multi-component Jaulent-Miodeck equation hierarchy, multi-component 2+1 dimensional GJ equation hierarchy and thecoupling Dirac equation hierarchy are considered. The R-matrix of coupling equation hierarchy isobtained through a generalized matrix spectral problem. For example, the R-matrix of couplingAKNS equation hierarchy is given.
Chapter 4 investigates the discrete nonisospectral problem. First, basing on loop algebraA1, a new sub-algebra is presented. The isospectral and nonisospectral Lax integrable couplingequation hierarchies are worked out. Second, the 2+1-dimension nonisospectral integrable cou-pling model is presented, the nonisospectral integrable coupling system of Blaszak-Marciniaklattice hierarchy is obtained under the nonisospectral condition of spectral parameterλ. TheA.R. Bishop Editor-in-Chief of《Physics Letters A》appraised that " The method gives twokinds of classification to a soliton equation, it is an interesting and important work ". Darbouxtransformation is considered. Last, the relation between the discrete soliton equation and the AKNS soliton equation hierarchy is given through the transform of potential functions.
Chapter 5, the fractional Hamiltonian structure of soliton equation is considered. Thefractional zero curvature equation is constructed by using of fractional differential formula, andobtain the fractional AKNS and fractional C-KdV equation by making use of the fractional zerocurvature equation, then their Hamiltonian structures are worked out. The fractional Poissonbracket is defined, a Hamiltonian system of fractional form is presented.
引文
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