Stability Of The Equilibrium Of The Infinite Dimensional Hamiltonian Systems

Many problems in physics and engineering are Hamiltonian partial differ-ential equations,the study on the stability of its coherent states(steady states,traveling waves,standing waves,etc.)can be converted into a study based on the linearized partial differential equation(?).Index theorem is an im-portant tool to study stability,in many cases,J has a bounded inverse,and there are many researches in this field.However,in many important Hamilto-nian partial differential equations(such as KDV equation,2-D Euler equation,etc.),J just has no bounded inverse.This paper summarizes the results of Lin Zhiwu and Zeng Chongchun,only assuming that J is an anti-self-dual opera-tor,and generalizes the previous results.Taking the 2-D Euler equation as an example,the index formula at the relative equilibrium is given.This paper is divided into four chapters:the first chapter introduces the background and development of the stability theory of equilibrium point of Hamiltonian system;In chapter 2,the symmetry of ?(JL)in linear Hamilto-nian system(?)are given in finite dimensional space,and the index formula in finite dimensional space is given.In chapter 3,the direct sum de-composition of phase space X,the exponential trichotomy underetJL,and an index formula linking the dimensions of generalized characteristic subspace of n-(L)and JL are summarized in infinite dimensional space.In chapter 4,taking the shear flow of 2-D Euler equation as an example,some conclusions about the stability at the relative equilibrium point are given by using the index formula.