The Research On Quasi-periodic Dynamics For Three Classes Of Differential Equations

In this thesis,we mainly use KAM theory to prove the existence and stability of quasi-periodic solutions for three classes of systems(finite-dimensional dynamical system,fully nonlinear Boussinesq system and quasi-linear Boussinesq system).KAM theory is named by Kolmogorov,Arnold and Moser,which is for studying the quasi-periodic solutions of almost integrable conservative systems.Since strict conditions of KAM theory,many problems encountered in practical application can not be directly applied to this theory.People hope to further develop and improve the KAM theory so that it can be applied to a wider range.Any finite dimensional embedded invariant torus of a Hamiltonian system,densely filled by quasi-periodic solutions,is isotropic.This property allows us to construct a set of symplectic coordinates in a neighborhood of the torus in which the Hamiltonian is in a generalized KAM normal form with angle-dependent coefficients.Based on this observation we develop an approach to KAM theory via a Nash-Moser implicit function iterative theorem.The key point is to construct an approximate right inverse of the differential operator associated to the linearized Hamiltonian system at each approximate quasi-periodic solution.The construction of an approximate inverse is thus reduced to solving a quasi-periodically forced linear differential equation in the normal variables.Applications of this procedure allow to prove the existence of finite dimensional Diophantine invariant tori of autonomous PDEs.Later,the scheme was also extended to reversible cases.We apply this technique to the research for three classes of differential equations.In this thesis,we first consider the linear quasi-periodic system Letting s0=(d+1)/2 and β=6n2+6τ-2,s≥s0 is sufficiently small,we prove that under the givea Hs0+β norm and the frequecy satisfying the Diopuantine condition,there exists a Cantor set with almost full Lebesgue measure and a quasi-periodic transformation of the form θ=θ,x=eP(θ)y with P(θ)∈Hs,which reduces above system into a constant system θ=ω,y=A*y where A*is a constant matrix close to A.Different from classical smooth results,our result requires smallness conditions only on a fixed low Sobolev norm(Hs0+β-norm)of the first perturbation.It is worth mentioning that our system does not need second Melnikov’s condition explicitly.As an application,we apply our results to smooth quasi-periodic Schr?dinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.Second,we consider a class of quasi-periodically forced perturbations of the dissipative Boussinesq systems with an elliptic fixed point in two cases:Hamiltonian case and reversible case.We prove the existence and linear stability of quasi-periodic solutions for the system with periodic boundary conditions.The method of proof is based on a NashMoser iterative scheme in the scale of Sobolev spaces.We not only study the Boussinesq systems in the Hamiltonian case and reversible case,but also solve the reduction problem of linearized operators with anti-symmetric coefficients.Third,the goal of this part is to develop a KAM theory close to an elliptic fixed point for quasi-linear perturbations of the dissipative Boussinesq systems with periodic boundary conditions.It is proved that the system admits small-amplitude quasi-periodic solutions corresponding to linearly stable d-dimensional elliptic diophantine invariant tori of an associated infinite dimensional Hamiltonian system.The method of the proof is based on a weak Birkhoff normal form algorithm and a nonlinear Nash-Moser iterative scheme.The analysis of the linearized operators at each step of the iteration is achieved by pseudo-differential operator techniques and a linear KAM reducibility scheme.The specific arrangement of this thesis is as follows:In Chapter 1,we give the preliminary knowledge that will be used in the following paper,such as some definitions,lemmas,propositions,etc.Then,we introduce the classical KAM theory and infinite dimensional KAM theory.Finally,we introduce the background and current status of our research,and give the main work we have done in this article.In Chapter 2,we give a detailed proof of reducibility under smooth topology in any matrix Lie subalgebra g for a kind of linear quasi-periodic system.In Chapter 3,we give a detailed construction of quasi-periodic solutions for nonlinear forced perturbations of dissipative Boussinesq systems in two cases:Hamiltonian case and reversible case.In Chapter 3,we give a detailed introduction of how to perform a weak Birkhoff normal form and the nonlinear Nash-Moser iterative scheme to construct the Sobolev quasi-periodic solutions for quasi-linear perturbations of dissipative Boussinesq systems.In the appendix,the relevant calculations and proofs mentioned in the text but not described in detail are supplemented.