On scattering for the quintic defocusing nonlinear Schrödinger equation on \(\mathbb{R}\times \mathbb{T}^{2}\) (Q2875807)

Motivated by the question how the geometry of the domain influences the asymptotic behavior of large solutions to nonlinear dispersive equations, the authors consider the quintic nonlinear Schrödinger equation on \(\mathbb{R} \times \mathbb{T}^2\), namely \((i \partial_t + \Delta_{\mathbb{R} \times \mathbb{T}^2})(u)=|u|^4u,\;\;u(t=0)=u_0 \in H^1(\mathbb{R} \times \mathbb{T}^2)\). In studying the effect of the domain, a natural direction is to investigate the phenomenon of scattering in which all nonlinear solutions asymptotically resemble linear solutions. One should note that, in the case of this equation, the domain's genus, and dimension, makes this case critical both at the level of energy and mass. The paper's first main result states that small initial data lead to solutions that are global and scatter. The second main result concludes global regularity of solutions of finite energy and, conditional to some extra assumption, large data scattering. The authors developed in their study an impressive technical machinery including global Strichartz and a delicate analysis of profile decomposition of the initial large-scale data.