Integrable Coupling System And Fractional Hamiltonian Structure Of Soliton Equation Hierarchy

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.