Infinite-dimensional KAM Theory And Its Application To Partial Differential Equations

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 the variable matrix, we establish an estimate of the solutions for the basic equation of multiple normal frequencies, non-critical unbounded KAM theory, that is, the small-divisor equation of higher order with large variable coefficients. Our estimate extends Kuksin's lemma for the estimate of the solutions for the scalar small-divisor equation with large variable coefficients.In Chapter 3, by using our estimate, we establish an abstract infinite dimensional KAM theorem dealing with multiple normal frequencies and non-critical unbounded perturbation vector-field. This theorem enlarges KAM theory and makes it more appli-cable to deal with partial differential equations.In Chapter 4, the KAM theorem in Chapter 3 is applied to the coupled KdV equa-tion which lies outside the validity range of all previous KAM theorems, so KAM tori and periodic solutions are obtained for it.In Chapter 5, by searching the partial Birkhoff normal form of order 10 and using a modified KAM theorem for infinite dimensional Hamilton system, we obtain the exis-tence and linear stability of quasi-periodic solutions for the wave equation with quintic term.