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We use modified scattering theory to demonstrate that small-data solutions tothe cubic nonlinear Schr\"odinger equation on rescaled waveguide manifolds,$\mathbb{R} \times \mathbb{T}^d$ for $d\geq 2$, demonstrate boundedness ofSobolev norms as well as weak instability.

Modified Scattering of Cubic Nonlinear Schr\"odinger Equation on Rescaled Waveguide Manifolds