| In Chapter1, we introduce the historical background, some recent results of KAM theory obtained in the literature and our main work in this paper.In Chapter2, it is proved that for a prescribed potential V(x) there are many quasi-periodic solutions of derivative nonlinear Schrodinger equation subject to Dirichlet boundary condition by means of a KAM theorem to a unbounded re-versible system.In Chapter3, we consider the non-autonomous Benjamin-Ono equation under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and partial Birkhoff normal form, we will prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.In Chapter4, we consider the d-dimensional beam equation under periodic boundary conditions: where is a real analytic function with V(α) real. We will apply the KAM Theorem in [26] into this system and obtain that for sufficiently small ε, there is a large subset S’of S such that for all s∈S’ the solution u of the unperturbed system persists as a time-quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients. |
