  We consider the cubic nonlinear Schr\"odinger equation with harmonic trapping
on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions
are trapped (a.k.a "cigar-shaped" trap), following the approach of
Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct
modified wave operators for small initial and final data respectively. The
asymptotic behavior turns out to be a rather vigorous departure from linear
scattering and is dictated by the resonant system of the NLS equation with full
trapping on $\mathbb{R}^{D-1}$. In the physical dimension $D=3$, this system
turns out to be exactly the (CR) equation derived and studied by
Faou-Germain-Hani. The special dynamics of the latter equation, combined with
the above modified scattering results, allow to justify and extend some
physical approximations in the theory of Bose-Einstein condensates in
cigar-shaped traps.
