Dimension reduction for Nonlinear Schrödinger equations

We discuss mathematical methods to derive Nonlinear Schrödinger 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 Schrödinger–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.