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

Asymptotic behavior of the nonlinear Schr\"odinger equation with harmonic trapping