Global Well-posedness For The Coupled KdV-Schr?dinger Equations And The Generalized Zakharov Equations

This thesis related to three classes of nonlinear evolution equations: Schr?dinger equation, Kd V equation and Zakharov equations, which are the famous mathematical model. The thesis studies the global well-posedness for the coupled nonlinear Kd V-Schr?dinger equations in dimensions one and generalized Zakharov equations in dimensions less than two.In Chapter 1, we introduce the background of our research, the context of this thesis and some preliminaries.Chapter 2 concerns with the existence and uniqueness for the coupled nonlinear Kd V-Schr?dinger equations in dimension one. The global existence of this equation through a modified equations is obtained. First, we give a priori estimates for the modified equations. Next, we prove the existence of the local solution for the modified equations by constructing successive approximation sequences. In view of part one and two, the existence and uniqueness of global so lution for the initial value problem of the modified equations have obtained. Then we can get my result easily.Chapter 3 concerns with the well-posedness for the classes of generalized Zakharov equations in dimensions less than two, which are solved by using a priori estimates and Galerkin method. It is classical and standard to obtain the existence of the local solution by Galerkin method, and we abbreviation the process of proof. Next we only get out a priori estimates for the equations in dimensions less than two in the next two parts respectively.