  We prove local and global well-posedness results for the Gabitov-Turitsyn or
dispersion managed nonlinear Schr\"odinger equation with a large class of
nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and
$H^1(\mathbb{R})$. Moreover, when the average dispersion is non-negative, we
show that the set of ground states is orbitally stable. This covers the case of
non-saturated and saturated nonlinear polarizations and yields, for saturated
nonlinearities, the first proof of orbital stability.
