Modified Scattering of Cubic Nonlinear Schrödinger Equation on Rescaled Waveguide Manifolds

We use modified scattering theory to demonstrate that small-data solutions to the cubic nonlinear Schrödinger equation on rescaled waveguide manifolds, $\mathbb{R}\times {\mathbb{T}}^d$ for $d\ge 2$, demonstrate boundedness of Sobolev norms as well as weak instability.