Re-implement Of Recursion Operators And Redesign Of Infinite-Dimensional Hamiltonian Operators

The Infinite-dimensional Hamiltonian system is a kind of partial differential equation(s)with special structure,which is widely used in various fields such as mathematical science,celestial mechanics and engineering mechanics.In infinite-dimensional Hamiltonian system,the article is mainly carried out the research on the re-implementation of recursion operators and the general construction of matrix Hamiltonian operators.In the first chapter,the article firstly introduces research problems;Secondly,after reading the literature,the research methods of recursion operators and the formal summary of infinite-dimensional Hamiltonian operators are roughly understand;Finally,the research ideas and main results of this article are explained.In the second chapter,this article is mainly discussed the structure of recursion operators and the solution of its coefficients under the infinite-dimensional linear Hamiltonian canonical system.With the help of the constant coefficient Hamiltonian operators,it is rewritten into a finite sum form,and applying the method of obtaining the general system recursion operators to infinite-dimensional linear Hamiltonian canonical system.By solving the corresponding deterministic equations,the general structure of the recursion operators allowed by the next-order constant coefficient Hamiltonian operators under constraint condition and the specific form of its coefficient are obtained.The conclusion is verified by an example in the correctness and convenience.In the third chapter,we consider the general design of infinite-dimensional matrix Hamiltonian operators and operator pairs of basing on the previous studies.Through constructing three forms of operators,the language of differential forms are used to prove the condition that Hamiltonian operators satisfy.Based on these conditions,a new matrix Hamiltonian operators are obtained and the conclusion is verified.After that,the construction of a new matrix Hamiltonian operator pairs are realized.Finally,a general summary is put concluded for the work done,and the short-comings of this article are pointed out,as well as the direction of work which would be possibly carried out in the future.