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We consider the focusing cubic nonlinear Schr\"odinger equation\begin{align}\label{CNLSS} i\partial_t U+\Delta U=-|U|^2U\quad\text{on$\mathbb{R}^2\times\mathbb{T}$}.\tag{3NLS} \end{align} Different from the 3DEuclidean case, the \eqref{CNLSS} is mass-critical and non-scale-invariant onthe waveguide manifold $\mathbb{R}^2\times\mathbb{T}$, hence the underlyinganalysis becomes more subtle and challenging. We formulate thresholds using the2D Euclidean ground state of the focusing cubic NLS and show that solutions of\eqref{CNLSS} lying below the thresholds are global and scattering in time. Theproof relies on several new established Gagliardo-Nirenberg inequalities, whosebest constants are formulated in term of the 2D Euclidean ground state. It isalso worth noting the interesting fact that the thresholds for globalwell-posedness and scattering do not coincide. To the author's knowledge, thispaper also gives the first large data scattering result for focusing NLS onproduct spaces.

Large data global well-posedness and scattering for the focusing cubic nonlinear Schr\"odinger equation on $\mathbb{R}^2\times\mathbb{T}$