  In this paper, we consider exponential integrators for semilinear Poisson
systems. Two types of exponential integrators are constructed, one preserves
the Poisson structure, and the other preserves energy. Numerical experiments
for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs
are presented. These geometric exponential integrators exhibit better long time
stability properties as compared to non-geometric integrators, and are
computationally more efficient than traditional symplectic integrators and
energy-preserving methods based on the discrete gradient method.
