Optimal convergence and long-time conservation of exponential integration for Schr\"{o}dinger equations in a normal or highly oscillatory regime

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schr\"{o}dinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of actions, momentum and density will be formulated and analysed. To this end, we derive continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. It is shown that these integrators exhibit optimal convergence and have near conservations of actions, momentum and density 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 presented.