On scattering for the defocusing quintic nonlinear Schr\"odinger equation on the two-dimensional cylinder

In this article, we prove the scattering for the quintic defocusing nonlinear Schr"odinger equation on cylinder $\mathbb{R}\times \mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L_x^2{h}^{\alpha }$, $0 < \alpha \le 1$, motivated by the linear profile decomposition of the mass-critical Schr"odinger equation in $L^2({\mathbb{R}}^d)$, $d\ge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schr"odinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schr"odinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schr"odinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].