Global Attractor For Two Types Of Semi-discrete Nonlinear Equations

This paper is devoted to the large time behavior and especially to the regular-ity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize a class of system of nonlinear Schrodinger-BBM equations and kawahara equation. We first prove that Schrodinger-BBM equations provide 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-ε×H2 and has a finite fractal dimension. Finally, we prove that the Kawahara equation possesses a global attractor in H5. The paper contains three parts as following:The first chapter mainly introduces the background of nonlinear Schrodinger-BBM equations and Kawahara equation, the basic theory of infinite dimensional dynamical system and method innovation.In the second chapter, we investigate the global attractor for a semi-discrete a class of system of nonlinear Schrodinger-BBM equations.In the third chapter, we study the global attractor for a semi-discrete five-order nonlinear Kawachara equation...