Infinite Dimensional Hamiltonian Method Of Nonlinear Evolution Equations

Infinite dimensional Hamiltonian system is a partial differential equation(s) with specific structure, it plays a significant role in the nonlinear science. In this dissertation, we take nonlinear evolution equations as research subject. By using of generalized Poisson bracket as its definition, we investigate the Hamiltonian intergrability and infinite dimensional Hamiltonian formalization from bi-Hamiltonian structure and Hamiltonian linear canonical expression. The main work not only involves the proof of intergrability of a new infinite dimensional Hamiltonian system via Magri theory, but also includes obtaining an efficient mechanization algorithm for realizing some kinds of Hamiltonian canonical formularize. We introduce the theory 'AC=BD' into Hamiltonian system, too.There are five chapters in this dissertation, main contents are described as following:Chapter 1: Firstly, by reviewing the shift of Hamiltonian system, such as from symplectic manifold to Possion form, and by summarizing three changes of definition, we realize that Hamiltonian equation(s) is completely decided by the agreed Poisson bracket and give a classification of Hamiltonian framework; Secondly, through the review of symplectic algorithm, operator spectrum theory and Hamiltonization and others problems of infinite dimensional Hamiltonian system, we elucidate that Hamiltonian method is a kind of important tool for the resolving some nonlinear evolution equations. At last we present the research background of this dissertation via the relation between the Hamiltonian structure and intergrable system.Chapter 2: We describe the related theory and concepts about the Hamiltonian operator. By using of Magri's bi-Hamiltonian theory, we prove completely the Hamiltonian intergrability of a third order nonlinear evolution equation. The key points in the proving processes include identification of a novel Muria transformation and determination of hereditary property for a recursion operator obtained by S.Yu.Sakovich. Through those results and a symplectic-cosymplectic decomposition, we show the Hamiltonian structure of this third order nonlinear evolution equation and other interrelated attributes, such as its hierarchy, conserved law, etc. Also, We discuss a specific Hamiltonian operator.Chapter 3: Utilizing the B.Fuchssteiner and A.S.Fokas theory, analysis of the constructing methods of multi-Hamiltonian system, we prove that the third order equation in chapter 2 possess tri-Hamiltonian structure. Accordingly, we obtain the third new conserved law of the equation and provide the sample of determining possible conserved law through constructing Hamiltonian structure.Chapter 4: We discuss the applications of mathematic mechanization for the expression of Hamiltonian canonical '(?)/(?)x'-pattern. Find a breakthrough based on mul-tivariate polynomial matrix by the decomposition of partial differential operator, which is one of approaches to resolve the inverse problem. Through these we can provide a concise algebra method, which can transfer inverse problem into resolving Hamiltonian canonical solution of a fixed operator equation, we also provide calculation processes and results of some equations in terms of this method.Chapter 5: In order to provide wider applications of the algorithm obtained by us, we pioneer a work which introduce the extended 'AC=BD' into Hamiltonian system.