Global Attractor For A Semi-discrete A Class Of System Of Nonlinear Schr(o|")dinger Equations

This paper is devoted to the large time behavior and especially to the regu-larity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize a class of system of nonlinear Schrodinger equations on R1 × R1. We first prove that such a semi-discrete system provides a discrete infinite dimensional dynamical system in H1 × H1 that possesses a global attractor Aτ in H1 × H1. We also show that the global attractor Aτ is regular, i.e. Aτ is actually included, bounded and compact in H3/2 × H3/2-ε and has a finite fractal dimension. the paper contains three parts as following:The first chapter is devote to the summary of the dissertation. First of all, we present the background of a class of system of nonlinear Schrodinger equations and basic theory of infinite dimensional dynamical system. Secondly, we present our goal and the main results of the paper. Thirdly, we describe the innovations and methods of the dissertation.In the second chapter, we investigate the long-time behavior of the semi-discrete a class of system of nonlinear Schrodinger equations.In the third chapter, we study the semi-discrete a class of system of nonlinear Schrodinger equations possess a global attractor and has a finite fractal dimension.