Modified scattering for the cubic Schr{\"o}dinger equation on product spaces: the nonresonant case

We consider the cubic nonlinear Schr{"o}dinger equation on the spatial domain $\mathbb{R}\times {\mathbb{T}}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani-Pausader-Tzvetkov-Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant.