  Parareal is a well-known parallel-in-time algorithm that combines a coarse
and fine propagator within a parallel iteration. It allows for large-scale
parallelism that leads to significantly reduced computational time compared to
serial time-stepping methods. However, like many parallel-in-time methods it
can fail to converge when applied to non-diffusive equations such as hyperbolic
systems or dispersive nonlinear wave equations. This paper explores the use of
exponential integrators within the Parareal iteration. Exponential integrators
are particularly interesting candidates for Parareal because of their ability
to resolve fast-moving waves, even at the large stepsizes used by coarse
propagators. This work begins with an introduction to exponential Parareal
integrators followed by several motivating numerical experiments involving the
nonlinear Schr\"odinger equation. These experiments are then analyzed using
linear analysis that approximates the stability and convergence properties of
the exponential Parareal iteration on nonlinear problems. The paper concludes
with two additional numerical experiments involving the dispersive
Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation.
These experiments demonstrate that exponential Parareal methods offer improved
time-to-solution compared to serial exponential integrators when solving
certain non-diffusive equations.
