  In 2015 Berezhiani & Khoury proposed a Superfluid Dark Matter (SFDM) model
where dark matter condenses and forms a superfluid on galactic scales. In the
superfluid state phonons interact with baryons, resulting in a behavior similar
to that of Modified Newtonian Dynamics (MOND). If one assumes that the DM
condensate rotates along with the galaxy, a grid of vortices should form
throughout the superfluid component if the rotation is fast enough. We aim to
investigate the size and impact of the vortices on surrounding baryons, and to
further investigate the parameter space of the model. We also look for a
possible vortex solution of the Lagrangian presented for the SFDM theory. We
first take a simple approach and investigate vortex properties in a constant
density DM halo, applying knowledge from condensed matter physics. We then use
the zero-temperature condensate density profile as a template to vary the DM
particle mass and the energy scale, $\Lambda$, of the SFDM model. Further, we
attempt to find a vortex solution of the theory by extracting the
Euler-Lagrange equation with respect to the modulus of the condensate
wavefunction from the full relativistic SFDM Lagrangian. For the constant
density approach we find that the vortices are on millimeter scale, and
separated by distances $\sim0.002\,\rm{AU}$. The parameter space of the model
is found to be substantial and a reduction in the DM particle mass leads to
larger vortices with a higher energy. However, none of the parameter
combinations explored here give both realistic values of $\Lambda$ and vortices
energetic enough to have an observational impact on the galaxy as a whole. The
vortex equation extracted from the Lagrangian of the model is unstable, and no
solution exhibiting the standard properties of a vortex solution is found.
