Random numerical stability of attractors for nonlinear Schrodinger equations on infinite lattices

We study both numerical and random approximations of the global attractor for the nonlinear Schrodinger lattice equation. First, for a discrete-time Schrodinger lattice equation via the implicit Euler scheme, we prove the existence and upper semicontinuity of numerical attractors towards the global attractor when the time-size tends to zero, while a new Taylor expansion in the Hilbert space plays key role in the proof. Second, we consider the random Schrodinger lattice equation and prove the upper semicontinuity from the random attractor to the global attractor when the noise-coefficient goes to zero. Third, we show the optimal bounds and lower semicontinuity of the three (global, numerical, random) attractors with respect to the external force and dissipative constant, respectively.