Quasi-periodic Solutions Of Reversible Schr(?)dinger Equations

The Schr(?)dinger equations are basic equations in quantum theory and can be used to describe the motion of microscopic particles. The study of all kinds of so-lutions and their properties of the Schr(?)dinger equations is one of the hottest topics in pure and applied mathematics, and there are a variety of research methods. From the point of view of dynamical systems, a lot of evolutionary Schr(?)dinger equations can be viewed as infinite-dimensional dynamical systems. In dynamical systems, we are very concerned with the existences and stability of some special solutions (equilibrium solutions, periodic solutions etc.). By study-ing these special solutions, we can know the behaviors of other solutions around them. Quasi-periodic solutions are also a kind of very important special solu-tions. And KAM (Kolmogorov-Arnold-Moser) theory is a very powerful tool for studying quasi-periodic solutions.Recently, in the study of KAM theory for partial differential equations (PDEs), higher dimensional and derivative nonlinear PDEs are two classes of PDEs with which people are more concerned. And we focus on the latter in this dissertation. In the study of KAM theory for derivative nonlinear PDEs, the main difficulty is that the derivatives in nonlinearity cause the loss of regularity of the correspond-ing perturbed vector field (or the operator defined by perturbed vector field is unbounded). Therefore we have to solve the variable coefficient homological equa-tions in the KAM proof, and S.Kuksin first studied such equations and obtained their estimates (called Kuksin’s lemma now).Based on this lemma, S.Kuksin, T.Kappeler and J.Poschel investigated quasi-periodic solutions of KdV-type equa-tions, respectively. However, Kuksin’s lemma can’t be applied to derivative non-linear Schr(?)dinger equations since the unboundedness of perturbed vector field is stronger. In 2011, using a generalized Kuksin’s lemma, J. Liu and X. Yuan proved a new KAM theorem and thus obtained smooth (C∞) quasi-periodic solutions for this class of Schr(?)dinger equations with Dirichlet boundary conditions. In 2014, they generalized this result to periodic boundary conditions. In 2012, J. Geng and J. Wu also studied this class of Schr(?)dinger equations (periodic boundary conditions), and they obtained linearly stable analytic quasi-periodic solutions with two Diophantine frequencies. In 2013, M. Berti, L.Biasco and M.Procesi studied quasi-periodic solutions for a class of Hamiltonian derivative wave equa-tions. Above-mentioned equations are all Hamiltonian. And more recently, there are some works on quasi-periodic solutions for reversible PDEs via KAM theory. In 2011, J. Zhang, M. Gao and X.Yuan built an infinite-dimensional reversible KAM theorem and obtained smooth quasi-periodic solutions for the reversible Schr(?)dinger equations with Dirichlet boundary conditions. In 2014, M. Berti, L.Biasco and M.Procesi proved a reversible KAM theorem that can be applied to reversible derivative wave equations.In this dissertation, we investigate quasi-periodic solutions of some classes of reversible derivative nonlinear Schr(?)dinger equations via KAM theory. There are three parts in our main work. In the first part, we first prove an infinite-dimensional reversible KAM theorem, and thus obtain smooth quasi-periodic solutions for the reversible derivative Schr(?)dinger equations with periodic bound-ary conditions which is the extension of work of J.Zhang, M. Gao and X.Yuan in 2011. In the periodic boundary case, normal frequencies in KAM theorem are double such that small divisor problems are more complex. And we overcome this by adding a commutativity condition to perturbed vector field.In the second part, using KAM method, we prove the existence and linear stability of analytic quasi-periodic solutions of the reversible derivative nonlinear Schr(?)dinger equations with higher order nonlinearities. Note that, in the first part, the time quasi-periodic solutions obtained are smooth. On the basis of work in the first part, we find another commutativity condition of perturbed vector field. And thus we build a new KAM theorem for infinite-dimensional reversible systems with two tangential frequencies and obtain a lot of linearly stable analytic quasi-periodic solutions with two Diophantine frequencies. Since the nonlinearities in equations are higher order, we only obtain a partial Birkhoff normal form and the choice of index set and analysis of Birkhoff normal form are more elaborate and complex.We also investigate the existence of invariant tori (thus quasi-periodic so-lutions) for a class of quasi-periodically forced reversible Schr(?)dinger equations with periodic and Dirichlet boundary conditions, respectively. Due to quasi-periodically forced perturbation, KAM theorems in the work by J. Zhang et al. and our first part can’t be directly applied to non autonomous form here, thus we give two improved KAM theorems in this part. In particular, in KAM theorem for the case of periodic boundary conditions, we introduce a new commutativity condition to perturbed vector field which is different from that in the first part.The dissertation is organized as follows.Chapter 1 is divided into four sections. And in the first two sections, we introduce some basics on Hamiltonian systems, reversible systems and KAM the-ory. In Section 3, we introduce the research background, methods and progress of quasi-periodic solutions for PDEs. In Section 4, we give an introduction to our main work.In Chapter 2, we study quasi-periodic solutions for a class of reversible deriva-tive nonlinear Schr(?)dinger equations with periodic boundary conditions. In Sec-tion 1, we give main result. Section 2 is a preliminary section in which we give some definitions and notations and study the properties of Lie bracket of vector fields and reversible vector fields. An infinite-dimensional reversible KAM theo-rem is presented in Section 3. In Sections 4 and 5, we prove this KAM theorem in detail. In Section 6, we apply the KAM theorem to prove main result. To start with, we rewrite the associated reversible vector field of Schr(?)dinger equa-tions in infinitely many coordinates and establish its regularity. Secondly, it is transformed into a Birkhoff normal form. Finally we verify the new vector field satisfies the assumptions in KAM theorem and complete the proof of main result.In Chapter 3, we consider a class of reversible derivative nonlinear Schr(?)dinger equations with higher order nonlinearities, and there are eight sections in this chapter. In Section 1, we give main result. Section 2 is a preliminary section in which we introduce a new commutativity condition to perturbed vector field. We devote Section 3 to the proof of main result, the proof is based on the Birkhoff normal form techniques and KAM theorem in Sections 4. And in Sections 5-8, we give the proof of KAM theorem.In Chapter 4, we consider a class of quasi-periodically forced reversible derivative nonlinear Schr(?)dinger equations. Similar to Chapters 2 and 3, we give main results and preliminary in Sections 1 and 2, respectively. In Section 3, we rewrite the Schr(?)dinger equations with both periodic and Dirichlet boundary conditions as infinite-dimensional reversible systems and investigate their Birkhoff normal forms. In Section 4, we prove main results, the proofs are based on two improved KAM theorems given in Section 5.Some necessary technical lemmas are listed in the final chapter of Appendix.