The Global Solutions For Several Classes Of Higher-order Nonlinear Schr(?)dinger Equation(System)

Schr(?)dinger type equation plays an important role in the field of physics.In this paper,we mainly use Banach's fixed point theorem to prove the existence and uniqueness of solutions.This paper is divided into three chapters.In Chapter 1,we study the global solutions for two kinds of 4m-th order energy-critical Schr(?)dinger equation.For the following nonlinear 4m-th order self-focusing Schr(?)dinger equation in the energy-critical case:(?)where ?1,?2 are non-zero real number,?2> 0,both u=u(x,t) and u0(x) are complex functions,and 8m/n<p<8m/n-4m is positive constant.We get the existence and the uniqueness of the global solution of the equation and the relationship between the solution and the initial value by using the Banach's fixed point theorem.For the following nonlinear 4m-th order non self-focusing Schr(?)dinger equation in the energy-critical case:(?)where ?1> 0,?2 <0 are real number,both u=u(x,t) and u0(x) are complex functions,and 0 <p<8m/n-4m.We get the existence and the uniqueness of the global solution of the equation and the relationship between the solution and the initial value by using the Banach's fixed point theorem under the condition that the initial value of the equation satisfies certain conditions.In Chapter 2,we study the existence and the uniqueness of the global solution for the following nonlinear 4m-th order Schr(?)dinger equation with exponential growth:(?)where m is a positive integer and ??{-1,0},both u=u(x,t) and ?(x) are complex functions,and f(u) satisfies the following:(?)By using the Banach's fixed point theorem,we prove the existence and the uniqueness of the global solution and the existence of the scattering operator.In Chapter 3,we study the global solution for the following nonlinear 2m-th order Schr(?)dinger equation system with exponential growth:(?)where a,b are real number and ?,?> 1,m is positive integer,u=u(x,t),v=v(x,t),?(x) and ?(x) are all complex functions.We prove the existence and the uniqueness of the global solutions of the above equations in Sobolev W1,p1{Rn)× W1,P2(Rn) space.We obtain the continuous dependence of the initial value and the decay estimate of the solution at the same time.In any dimension space,we also prove the existence and uniqueness of the global solution for the above higher-order nonlinear Schr(?)dinger equations in Sobolev Hs,P1(Rn)×Hs,p2(Rn) space,and get the continuous dependence on the initial value.