A cubic nonlinear Schrödinger equation (NLS) with periodically varying dispersion coefficient, as it arises in the context of fiber-optics communication, is considered. For sufficiently strong variation, corresponding to the so-called strong dispersion management regime, the equation possesses pulse-like solutions which evolve nearly periodically. This phenomenon is explained by constructing ground states for the averaged variational principle and justifying the averaging procedure. Furthermore, it is shown that in certain critical cases (e.g. quintic nonlinearity in one dimension and cubic nonlinearity in two dimensions) the dispersion management technique stabilizes the pulses which otherwise would be unstable. This observation seems to be new and is reminiscent of the well-known Kapitza’s effect of stabilizing the inverted pendulum by rapidly moving its pivot. © 2001 Published by Elsevier Science B.V.