With Weak Damping Two-dimensional Nonlinear Schr (?) Dinger Equation Finite Difference Scheme For The Time Behavior

The two-dimensional nonlinear Schrodinger equation with weakly damped that possesses a global attractor is considered. The existence of an attractor is one of the most important characteristics for a dissipative system, The long-time dynamics is completely determined by the attractor of the system. At present, most studies are linked to the semi-discrete scheme and the one-dimensional spatial variable.But there are very few studies about the full-discrete scheme and the higher dimensional spatial variable. Furthermore, the former can't be extended directly to the latter . As the result ,the latter is focused by the reseachers nowadays. The dynamical properties of the discrete dynamical system which is generated by a finite difference scheme are analysed,and the existence of a global attractor for the discrete dynamical system is proved . In order to prove the existence of an attractor , we first sets up the interpolation unequality in two-dimensional space, from which, the simutaneous relation of ||·||LP between ||·||L2 and ||·||1 is derived.Then, it constructs the finite difference scheme of the equation and from which establishs the functional En and its recursion relation in the space of H2-functions.Therefore,the existence of an absorbing set in the space of H2-functions is drawn.Finally , the existence of an attractor of the equation in two-dimensional space is obtained by using the conclusion in [3].