UC Berkeley

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"{o}dinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. We first prove the local well-posedness in $C(I;L_x^2H_y^s)\cap C(I;L_{x,y}^4)$ for solutions with initial data $u_0\in H^{0,1}_xL_y^2\cap L_x^2H_y^s$. Then, for sufficiently small, smooth, decaying data, we prove global existence and derive modified asymptotic dynamics by using the wave packet method and normal form corrections. The modified scattering behavior on $\mathbb{R}\times\mathbb{T}^d$ combines the modified scattering of the cubic NLS on real line $\mathbb{R}$ with cubic NLS dynamics on torus. We also consider the corresponding asymptotic completeness problem.