Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times

Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrödinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrödinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.