Kam Theory And Hamiltonian Partial Differential Equations

In this thesis, we are devoted to studying the existence of quasi periodic solutions and almost periodic solutions in HPDEs. We will mainly discuss semi-linear HPDEs and quasi-linear HPDEs. The main content of this thesis includes two important parts: the existence of almost periodic solutions for semi-linear quantum harmonic oscillators; the existence of quasi periodic solutions in quasi-linear HPDEs.In the first chapter, we will first briefly review the development of the KAM theo-ry. Then, we will give the classification for HPDEs according to the unboundedness of the perturbation. Finally, we will state the known results for almost periodic solutions in semi-linear HPDEs and quasi periodic solutions in quasi-linear HPDEs.In the second chapter, we will discuss the existence of almost periodic solutions for semi-linear quantum harmonic oscillators. The main method of solving this problem is an abstract infinite dimensional KAM theorem. It should be emphasized that the decay of Hermite function(hj(x))j≥is crucial for constructing almost periodic solu-tions.In the third chapter, we will deeply discuss the quasi-linear HPDEs. First, we will point out the difficulties of dealing with quasi-linear HPDEs. Then, we will give three related results:slow quasi-periodic linear systems are reducible; perturbed KdV equation admits real analytic quasi-periodic solutions with more Diophantine frequen-cies; the derivative nonlinear Schrodinger equation admits real analytic quasi-periodic solutions. In the fourth, fifth, sixth chapters, we will give the detailed proof of the three re-sults which were mentioned in the third chapter. For the first result, the proof is based on a key lemma and Newtonian iteration. For the second result, the proof is based on a conserved quantity, Toplitz-Lipschitz property and an abstract infinite dimensional KAM theorem. For the third result, the proof is based on a partial Birkhoff normal form reduction and an abstract infinite dimensional KAM theorem.