Invariant Manifold Of Traveling Wave Solutions Of A Generalized Boussinesq Equation

We study the existence of locally invariant manifolds for the following generalized Boussinesq equation(?),The model is widely used to describe wave propagation and other physical phenomena.the Boussinesq equation is linearized at the traveling wave solution and written into the Hamiltonian structure as follows.? t(u)=JL(u),u ∈La2(R).Considering that in the natural phase space La2(R),the essential spectrum of the JL operator of the Boussinesq equation in the central space cannot be used to construct manifolds.In order to construct as many invariant manifolds as possible,weight space La2(R)is selected in this thesis instead of natural phase L2(R)space.This paper firstly introduces some definitions and symbols that will be used in the elaboration and proof process,such as the definition and symbolic representation of discrete spectrum and essential spectrum,and proves that the eigenvalues of JL operator can be characterized by Evans function.In chapter 3,the spectral properties of the JL operator in the weight space La2(R)are analyzed precisely by using Evans function,and the exact positions of eigenvalues embedded in the essential spectrum are obtained.It is found that the essential spectrum of the JL operator in the central space moves to the left half complex plane,so that more spectrum can be reserved to construct the manifold.In chapter 4,according to the spectral properties of JL operators,the weight space is decomposed into the direct sum of stable space,unstable space and central space,and the concrete structure of Hamiltonian operators in the weight space is obtained based on space decomposition.In chapter 5,we prove the exponential dichotomies of semigroups on weight Spaces,which shows that there are nonlinear local invariant submanifolds near invariant sub spaces.Finally,the structure of stable,unstable,centrally stable and centrally unstable manifolds is prospected.