The Application Of KAM Theory In The Existence Of Quasi-periodic Solutions For PDEs

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.