  Dark matter as a Bose-Einstein condensate, such as the axionic scalar field
particles of String Theory, can explain the coldness of dark matter on large
scales. Pioneering simulations in this context predict a rich wave-like
structure, with a ground state soliton core in every galaxy surrounded by a
halo of excited states that interfere on the de Broglie scale. This de Broglie
scale is largest for low mass galaxies as momentum is lower, providing a simple
explanation for the wide cores of dwarf spheroidal galaxies. Here we extend
these "wave dark matter" ($\psi$DM) predictions to the newly discovered class
of "Ultra Diffuse Galaxies" (UDG) that resemble dwarf spheroidal galaxies but
with more extended stellar profiles. Currently the best studied example, DF44,
has a uniform velocity dispersion of $\simeq 33$km/s, extending to at least 3
kpc, that we show is reproduced by our $\psi$DM simulations with a soliton
radius of $\simeq 0.5$ kpc. In the $\psi$DM context, we show the relatively
flat dispersion profile of DF44 lies between massive galaxies with compact
dense solitons, as may be present in the Milky Way on a scale of 100pc and
lower mass galaxies where the velocity dispersion declines centrally within a
wide, low density soliton, like Antlia II, of radius 3 kpc.
