Numerical Methods for Some Nonlinear Schrödinger Equations in Soliton Management

In this work, we consider the numerical solutions of a dispersion-managed nonlinear Schrödinger equation (DM-NLS) and a nonlinearity-managed NLS equation (NM-NLS). The two equations arise from the soliton managements in optics and matter waves, and they involve temporal discontinuous coefficients with possible frequent jumps and stiffness which cause numerical difficulties. We analyze to see the order reduction problems of some popular traditional methods, and then we propose a class of exponential-type dispersion-map integrators for both DM-NLS and NM-NLS. The proposed methods are explicit, efficient under Fourier pseudospectral discretizations and second order accurate in time regardless the jumps/jump-period in the dispersion map. The extension to the fast & strong management regime of DM-NLS is made with uniform accuracy.