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In this paper, we formulate and analyse exponential integrations when appliedto nonlinear Schr\"{o}dinger equations in a normal or highly oscillatoryregime. A kind of exponential integrators with energy preservation, optimalconvergence and long time near conservations of actions, momentum and densitywill be formulated and analysed. To this end, we derive continuous-stageexponential integrators and show that the integrators can exactly preserve theenergy of Hamiltonian systems. Three practical energy-preserving integratorsare presented. It is shown that these integrators exhibit optimal convergenceand have near conservations of actions, momentum and density over long times. Anumerical experiment is carried out to support all the theoretical resultspresented in this paper. Some applications of the integrators to other kinds ofordinary/partial differential equations are also presented.

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