On scattering for the defocusing quintic nonlinear schrodinger equation on the two-dimensional cylinder

In this article, we prove global well-posedness and scattering for the defocusing quintic nonlinear Schrodinger equation on the cylinder R × T in H1. We establish an infinite vector-valued linear profile decomposition in L2 xhα , 0 < α ≤ 1, motivated by the linear profile decomposition of the mass-critical Schrodinger equation in L2(Rd), d ≥ 1. Then by using the solution of the one-discretecomponent quintic resonant nonlinear Schrodinger system, whose scattering can be proved by using the techniques established by Dodson, to approximate the nonlinear profile, we can prove scattering in H1 by using the concentration compactness/rigidity method. As a by-product of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrodinger system, we also prove the scattering conjecture for the two-discrete-component quintic resonant nonlinear Schrodinger system presented by Hani and Pausader in [Comm. Pure Appl. Math., 67 (2014), pp. 1466-1542].