  This paper investigates the feasibility of simulating Fuzzy Dark Matter (FDM)
with a reduced number of spatial dimensions. Our aim is to set up a realistic,
yet numerically inexpensive, toy model in $(1+1)$-dimensional space time, that
- under well controlled system conditions - is capable of realizing important
aspects of the full-fledged $(3+1)$-FDM phenomenology by means of
one-dimensional analogues. Based on the coupled, nonlinear and nonlocal
$(3+1)$-Schr\"odinger-Poisson equation under periodic boundary conditions, we
derive two distinct one-dimensional models that differ in their transversal
matter distribution and consequently in their nonlocal interaction along the
single dimension of interest. We show that these discrepancies change the
relaxation process of initial states as well as the asymptotic, i.e.,
thermalized and virialized, equilibrium state. Our investigation includes the
dynamical evolution of artificial initial conditions for non-expanding space,
as well as cosmological initial conditions in expanding space. The findings of
this work are relevant for the interpretation of numerical simulation data
modelling nonrelativistic fuzzy cold dark matter in reduced dimensions, in the
quest for testing such models and for possible laboratory implementations of
them.

Cosmology and Nongalactic Astrophysics;Quantum Gases;Pattern Formation and Solitons