  In this paper we consider the long time behavior of solutions to the cubic
nonlinear Schr\"odinger equation posed on the spatial domain
$\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small,
smooth, decaying data we prove global existence and derive modified asymptotic
dynamics. We also consider the asymptotic completeness problem.
