Exact and approximate moments of a propagating pulse

We have recently developed a phase space approach for studying dispersive wave propagation. Using this approach, we have derived a simple approximation method. We show that this approximation gives the exact low order moments of the wave for all time. In particular, the mean motion and spread of a pulse are exact. The approximation also gives the exact moments of the spatial spectrum of the wave, for all orders. We also consider local moments, and show that the low-order local mean and spread are exact. We argue that the reason why the approximation works well for all time is precisely because it preserves important moments of the wave. We compare these results with the moments of the stationary phase approximation, which are accurate only for large time.