  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.
