Kam Theory And Its Application In Critical Condition

In Chapter 1, we introduce the historical background, some recent results of KAM theory obtained in the literature and our main work in this paper.In Chapter 2, by the analysis of a Toplitz matrix and its exponential, we establish a new estimate of the solutions for the basic equation of unbounded KAM theory, that is, the small-divisor equation with large variable coefficients. Our estimate can be applied to both non critical case and critical case.In Chapter 3, by using our estimate, we prove a reduction theorem of KAM type including critical case, which entails the pure-point nature of the Floquet spectrum of the quantum Duffing oscillator with a small perturbation temporal quasi-periodic with non-resonant frequencies. This solves the problem proposed by Bambusi and Graffi in Commun. Math. Phys.219,465-480 (2001).In Chapter 4, by using our estimate, we establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field. This theorem extends Kuksin's non critical unbound KAM theorem to critical case, and consequently brings a large class of Hamiltonian PDEs containing the derivative (?) in the perturbation into the validity range of KAM theory.In Chapter 5, the KAM theorem in chapter 4 is applied to derivative nonlinear Schrodinger equation and perturbed Benjamin-Ono equation which lie outside the va-lidity range of all previous KAM theorems, so KAM tori and thus quasi-periodic solu-tions are obtained for them.