The Quasi-periodic Solutions Of Derivative Nonlinear Schrodinger Equations

In this dissertation, we consider nonlinear Schrodinger equations which have a wide range of applications in quantum mechanics. Since the infinite dimen-sional KAM theory has been set up, more and more authors began to study the existence of finite dimensional invariant tori(quasi-periodic solutions) and infinite dimensional invariant tori (almost periodic solutions) of the nonlinear Schrodinger equations as Hamiltonian systems.To integrable systems, the dynamics are clear. However, the integrable Sys-tems are few and more systems are nearly integrable in reality. Poincare saw the dynamics of nearly Hamiltonian systems as "Fundamental Problem of Dynamics" In 1950s to 1960s, the classical KAM theory was set up by three international cel-ebrated mathematician:A.N. Kolmogorov[64], V.I. Arnold[1] and J.K. Moser[93]. Because of its important value in application, KAM theory was seen as one of the high points of 20th century mathematics. In the late 1980s and early 1990s, the finite dimensional KAM theory was generalized and applied to Hamiltonian partial differential equations by S.B. Kuksin[66][67][68][69], W. Craig and C.E. Wayne[29] [30], and then they set up the infinite dimensional KAM theory. Later, a very explicit KAM-like theorem for PDEs was developed by J. Poschel[105] and was applied to nonlinear Schrodinger equations [70] and nonlinear wave equa-tions [106] under Dirichlet boundary conditions successfully, while J. Boungain [20] extended the concepts to more general PDEs. As all mentioned above are on the bounded perturbations, Poschel [59] generalized the KAM theorem for unbounded perturbation by using generalized normal forms and Kuksin’s lemma solving variant coefficient homological equations[71]. However, there still are a large class of important PDEs which can not be covered by above theorems, such as derivative nonlinear wave equations and derivative nonlinear Schrodinger equa-tions etc. Recently, Liu and Yuan[80] generalized Kuksin’s lemma and obtained the KAM theorem on limit case(see [81]). Then Liu and Yuan[81][82], Zhang, Gao and Yuan[132] considered several classes of derivative nonlinear Schrodinger equations, in which there are either Hamilton systems or reversible systems.Different form Dirichlet boundary conditions, above Poschel’s KAM the-orems are out of work under periodic boundary conditions as they give rise to double eigenvalues. For the existence of quasi-periodic solutions of nonlinear wave equations under periodic boundary conditions, the first result by KAM methods is due to Chierchia and You[27]. In fact, before them, Craig and Wayne[29] had already proved the existence of periodic solutions by Lyapunov-Schmidt reduc-tion method and techniques by Frohlich and Spencer. The method developed by Craig and Wayne[29][30] and improved by Bourgain[15][18][19] is called C-W-B method.For higher dimensional Hamiltonian PDEs, the difficulty is greater and the progress is slower. The earliest result is due to Bourgain[18] investigated the small amplitude periodic solutions for 2D Schrodinger equations. In this direction we refer to Bourgain[19], Geng and You [43], Eliasson and Kuksin[33]. Geng, Xu and You [46], Geng and You [49], Eliasson, Grebert and Kuksin[34][35] etc.There are three famous classes of derivative nonlinear Schrodinger equations in practical applications: In 2012, Geng and Wu [48] considered the derivative nonlinear Schrodinger Equa-tions under periodic boundary conditions: and obtained the real analytic quasi-periodic solutions with two frequencies. Meanwhile, Liu and Yuan [82] considered a similar equation under periodic bound-ary conditions: and got smooth quasi-periodic solutions with N frequencies. It should be pointed out that above two results are different as two different strategies they used. By introduced "Compact Form" and "Gauge Property", Geng and Wu reduced the variant coefficient homological equations to constant ones, then via KAM itera-tion obtained the real analytic quasi-periodic solutions. On the other hand, by using the improved lemma solving variant coefficient homological equations and assuming the perturbation satisfying special form, Liu and Yuan got the smooth quasi-periodic solutions via KAM iteration. Especially it should be noted that because of "Compact Form" and "Gauge Property", the number of frequencies of quasi-periodic solutions obtained by Geng and Wu is only two, while the num-ber is any positive integer in Liu and Yuan’s result. In this dissertation, we first consider the second class of derivative Schrodinger equations:Chen-Liu-Lee equation: under periodic boundary conditions by Geng and Wu’ method[48]. By "Com-pact Form" and "Gauge Property", we obtained the real analytic quasi-periodic solutions with two frequencies.For the perturbation with quasi-periodic forcing, the systems are nonau-tonomous ones. For completely resonant wave equations with periodic forcing, the periodic solutions are obtained by Rabinowitz using the variational method and the Lyapunov-Schmidt reduction. Later, combing with Lyapunov-Schmidt reduction and Nash-Moser iteration, Berti and Procesi[11] investigated the ex-istence of quasi-periodic solutions with two frequencies for completely resonant wave equations with periodic forcing. Then Jiao and Wang[54] proved the exis-tence of quasi-periodic solutions of nonlinear Schrodinger equations with quasi-periodic forcing via Birkhoff normal forms and KAM method. In recent years, Zhang and Si[133], Si[118] discussed the existence of quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing under Dirichlet bound-ary conditions and periodic boundary conditions respectively. Then Wang and Si[119], Rui and Si [109] considered the quasi-periodic solutions for the nonlinear beam equations and Schrodinger equations with quasi-periodic forcing respec-tively. It should be noted that all above perturbations are bounded. When the perturbations are unbounded, especially limit case, it is difficult even the sys-tems are autonomous. For the quasi-periodic forced unbounded perturbations, the results are few. First, Mi and Zhang[89] considered the Benjamin-Ono equa-tions with quasi-periodic forced perturbation via KAM method. Baldi, Berti and Montalto [7] considered linear Airy equations with quasi-linear perturbation by Nash-Moser iteration and KAM iteration. Using the same method, R. Feola and M. Procesi[39] discussed fully nonlinear forced reversible Schrodinger equations. In this paper, the author considers the derivative nonlinear Schrodinger equations with quasi-periodic forced perturbations: under periodic boundary conditions and proves the existence of invariant tori. Where B is a positive constant, g is real analytic and quasi-periodic on time t with frequency vector β=(β1,β2, ...,βm). The proof is based on Birkhoff normal forms and KAM iteration. It should be pointed out that under periodic boundary conditions, as the eigenval-ues are double, we can not use Poschel’s infinite dimensional KAM theorem for unbounded perturbations. Following Liu and Yuan[81][82], we consider general normal forms and suppose the perturbation satisfying the similar special form: perturbation P only consists of monomials where Then we proved the existence of invariant tori by solving variant coefficient ho-mological equations. We divided it into two cases:(1) β is any real vector; (2) β is co-linear with fixed β, that is β= λβ Rm, λ ∈ [1/2,3/2]. For the first case, we can use an improved Liu and Yuan’s KAM theorem satisfying special perturbations directly and get smooth quasi-periodic solutions which the number of frequencies is any positive integer. But for the second case, as the frequency of perturbation is "fixed", we do not have enough parameters and then the mea-sure estimate methods in [81] [82] can not be used in our problem. By different methods of measure estimate, we get smooth quasi-periodic solutions which the number of frequencies is any positive integer.This dissertation is divided into five chapters as following:In chapter one, we give the related theory on Hamiltonian systems and KAM theory. There are also four sections in this chapter. We introduce finite dimen-sional Hamilton systems theory in the first section, which consists of Hamilto-nian vector fields and transformation theory, integrable Hamiltonian systems and Birkhoff normal forms. In section 2, we mainly introduce classical KAM theory. Then we describe the three main parts in the study of Hamiltonian perturbation theory in section 3 briefly:Classical stability, Geometric stability and Instability. In the last section, we describe in detail the infinite dimensional KAM theory and applications in the study of Hamiltonian partial differential equations, especially nonlinear wave equations and nonlinear Schrodinger equations.In the second chapter, we list some basic definitions and conclusions in the application of KAM theory:such as quasi-periodic function, real analytic function and Cauchy estimate etc.In chapter three, we use Geng and Wu’s method [48] discuss the Chen-Liu-Lee equations under periodic boundary conditions: By "Compact Form" and "Gauge Property" we obtain the real analytic quasi-periodic solutions with two frequencies.In the forth and fifth chapter we study the derivative nonlinear Schrodinger equations with quasi-periodic forced perturbations and the existence of invariant tori under periodic boundary conditions. We con-sider this problem in the case where β is any real vector in chapter four and in the case where β is co-linear with fixed β in chapter five.