Study About The Recursion Operators And Inverse Problem Of Infinite Dimensional Hamiltonian System

Infinite dimensional Hamiltonian system is a kind of partial differential equation(s)with special structure,it is a basic form commonly used to solve practical problems such as physics,mechanics,control etc.In this paper,under the framework of Hamiltonian system,the recursion operator and the inverse problem is mainly discussed.Firstly,translate the system of general differential equations of recursion operator theory to the Hamiltonian system and thus a method for obtaining cyclic operators is obtained;Then the inverse problem of infinite dimensional Hamiltonian system is discusses in two points.In the first chapter,the paper firstly introduces the research object,describes the research direction of Hamiltonian system and lists some related definitions of infinite dimensional Hamiltonian system;Secondly,expounded research situation of recursion operator and the method of inverse problem are introduced;Finally,the main research ideas and work are introduced.In the second chapter,we explore the method of obtaining the recursion operator of Hamiltonian system,mainly to transplant a conclusion of recursion operator into the infinite dimensional Hamiltonian system,and thus obtain a general conclusion of recursion operator of Hamiltonian system with constant coefficients,the feasibility which is verified by some examples.In the third chapter,on the basis of previous studies,the inverse problem of Hamiltonian system is discussed in two points,first of all,based on the Taylor expansion,we get a improved calculation method of state variable,which is mainly reflected in the calculation of the decomposition factor Q;Second of all,we discuss the canonical form of high order equations which is determined by the operators having been discussed in papers,and by searching for transformation,we find the canonical form of equations determined by a class of new Hamiltonian operator.Finally,we make a brief summary of the work done,analyze the deficiency,and point out the direction of the next may work.