Researching On The Infinite Dlmensional Hamilton System Formalization And Its Operators

The infinite dimensional Hamilton system evolves from classical finite dimensional Hamilton system. Its purpose is to solve the problems of many partial differential equations in mathematics and mechanics. The properties of infinite dimensional Hamilton system in-clude Hamilton differential operator, recursion operator, conservation law, the distinguished functional, variational calculus inverse problem and so on. It is of great significance to study its properties. For example,Hamilton canonical form of the differential operator can sim-plify the equation;Recursion operator of differential equation can be used to produce more symmetry;Hamilton perturbation theory is studied easily under the regular form;The law of conservation plays an important role in studying the integrability of the nonlinear evolution equations;Distinguished functional can get laws of conservation of some Hamilton systems. This paper mainly studied the inverse problem of infinite dimensional Hamilton system, namely the implementation issues of canonical form, discussed the special Hamilton matrix differential operator, and obtained some meaningful conclusions.In the first chapter,the author first described the infinite dimensional Hamilton System and research status; Then, introduced the research progress and the research direction about the infinite dimensional Hamilton operator both at home and abroad,and gave the article re-search goal; Finally, there was the structure of the paper.In the second chapter, the author mainly presented an realization way of Hamilton canonical form, which is standardized on the basis of predecessors’standardization. In this part, the author first gave the concrete steps of translating equation or equations into a canon-ical form; Then, promoted the method that translate an evolution equation determined by the special kind of third order Hamilton operator into canonical form; In the end, further confirmed the effectiveness of the method by the validation of several examples.In the third chapter, several special kinds of Hamilton differential operators have been mainly discussed. Firstly, the author made a summary of scalar Hamilton operators in detail and promotion based on the existing materials; Secondly, some new results were obtained by the discussion of the second order and the third order matrix Hamilton operator; Finally, the paper proved that these couples of Hamilton operators are compatible.In the fourth chapter, the author not only made a summary and pointed out the prob-lems of this study, but also introduced the future study directions and the work we need to do.