The Well-posedness Study Of Several Nonlinear Schr(?)dinger Equations In Plasma

Nonlinear Schr(?)dinger equation is an importantly mathematical model in differ-ential physical field including nonlinear optics, plasma physics, quantum field theory and so on. It has been used plentifully, especially used in describing the refract of the electromagnetic wave in dielectric with certain field strength, the propagation of the magnetized plasma with a constant magnetic field and the Bose-Einstein conden-sates. In this paper, we mainly study the nonlinear Schr(?)dinger equations in atomic Bose-Einstein condensates and the nonlinear Schr(?)dinger equations with derivative, studying the posedness of the solution to these equations in periodic boundary condi-tions and the Cauchy boundary conditions.In Chapter 1, background knowledge. Firstly, we introduce the physical back-ground of the nonlinearly coupled Schr(?)dinger equations in atomic Bose-Einstein con-densates and the current research about these equations in mathematic. Secondly, we introduce the physical background of the nonlinearly coupled Schr(?)dinger equations with derivative and the research status.In Chapter 2, we prove the existence and uniqueness of globally smooth solution and blow-up phenomenon for a nonlinearly coupled Schr(?)dinger system in atomic Bose-Einstein condensates. Firstly, we are concerned with the periodic boundary problem. Applying matrix theory and the Banacch's fixed point theorem, we obtain the existence and uniqueness of locally smooth solution for the equations. Then combining with a priori estimates of the solution, the existence and uniqueness of globally smooth solution is obtained for the periodic boundary condition. Since the a priori estimates of the solution are independent on periodic L, let L ?? the existence and uniqueness of globally smooth solution for Cauchy problem is established. Finally, under some conditions of the equation's coefficients and the nonlinear term's exponent p, we get the blow-up theorem of the solution.In Chapter 3, we study the existence and uniqueness of globally smooth solution for the nonlinear Schr(?)dinger equations in atomic Bose-Einstein condensates in two dimension spaces. Since the Sobolev embedding H1?L? in two dimensional case is invalid, we can't directly apply Sobolev embedding theorem to obtain the existence of the global solution. We apply the logarithm Sobolev inequality to get a priori esti-mates. Then applying matrix theory and the Banacch's fixed point theorem, we obtain the existence and uniqueness of locally smooth solution for the equations in small ini-tial data conditions. Then combining with a priori estimates of the solution, we get the existence and uniqueness of globally smooth solution for the periodic boundary condition. Since the a priori estimates of the solution are independent on periodic L, let L ?? the existence and uniqueness of globally smooth solution for Cauchy problem in small initial data conditions is established.In Chapter 4, the fractional nonlinear Schr(?)dinger equations for atomic Bose-Einstein condensates are studied in one dimension space. Firstly, we apply compactness method to get the existence of global solution. Then combing the a priori estimates, the existence and uniqueness of global smooth solution are obtained with the periodic boundary conditions. Since all the a priori estimates are not concerned with domain ?, let L??, the existence and uniqueness of globally smooth solution for Cauchy problem is established.In Chapter 5, we concern on the nonlinear Schr(?)dinger equations with derivative. By using the Galerkin method and a priori estimates, we obtain the global existence of the weak solution in small initial data conditions.In Chapter 6, we study the Cauchy problem of the coupled Schr(?)dinger equations with derivative in one dimension space. Using the Fourier restriction norm method introduced by J. Bourgain, contraction mapping principle and maximum element estimation, we obtain the local well-posedness for initial data in HS(R) × Hs(R)(s> 1/2).