Small data scattering for the nonlinear Schrödinger equation on product spaces

Abstract

We consider the cubic nonlinear Schrödinger equation, posed on ${\mathbb{R}}^n\times M$, where $M$ is a compact Riemannian manifold and $n\ge 2$. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.

10 pages, slightly revised version, to appear on Comm. PDE