Modified scattering for the cubic Schrödinger equation small data solution on product space

Abstract

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times {\mathbb{T}}^d$, $1\le d\le 4$. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotic dynamics. We also consider the asymptotic completeness problem.

38 pages