Optimal Convergence and Long-Time conservation of Exponential Integration for Schrödinger Equations in a Normal or Highly Oscillatory Regime

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schrödinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of density, momentum and actions is formulated and analysed. To this end, we propose continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. We establish that these integrators exhibit optimal convergence and have near conservations of density, momentum and actions over long times. A numerical experiment is carried out to support all the theoretical results presented in this paper. Some applications of the integrators to other kinds of ordinary/partial differential equations are also discussed.