Dimension reduction for Nonlinear Schr\"odinger equations

Abstract

We discuss mathematical methods to derive Nonlinear Schr\"odinger equations (NLS) in "low dimensional" settings, i.e. the 3-dimensional physical space e.g. to 2 or 1 space dimensions. Beside from the case the system exhibits an internal symmetry we consider the approaches of dimension reduction via confinement limits and the method of variation. We deal with 2 types of NLS: nonlocal nonlinearities like the Hartree equation, including the Schr\"odinger--Poisson system (SPS), and local nonlinearities like the Gross--Pitaevskii equation (GPE). Our theoretical considerations of dimension reduction get finally illustrated by numerical examples in a "quasi 1-d" setting.