Study On Several Boundary Value Problems Of Fourth-Order Schr?dinger Equations

The fourth-order Schr?dinger equation:(?)(where a,b ∈ R and a≠0,f denotes nonlinear real function)is an important class of dispersive partial differential equations,which has important applications in quantum mechanics,non-linear optics and plasma physics,etc.In this thesis,we study the well-posedness and controllability of the initial-boundary value problem for fourth-order Schrodinger equations.The first chapter is the introduction,which introduces the background,research progress,research content and research results of the equation.In Chapter 2,we study the initial-boundary value problem for the fourth-order Schrodinger equation on a periodic domain.(?)By using the HUM principle,it is proved that the control system(0.1)is precisely controllable.In Chapter 3,we study the initial-boundary value problem of fourth-order Schrodinger equation with nonlinear boundary conditions on semi-bounded domain.(?)The local well-posedness of the initial-boundary value problem(0.2)is proved by using the Banach fixed point theorem.In Chapter 4,we study the initial-boundary value problem of two-dimensional fourthorder Schrodinger equation.(?)The local well-posedness of the initial-boundary value problem(0.3)is proved by using Banach fixed point theorem.