  In this note, we have addressed various issues on zero temperature
holographic superfluids. First, inspired by our numerical evidence for the
equality between the superfluid density and particle density, we provide an
elegant analytic proof for this equality by a boost trick. Second, using not
only the frequency domain analysis but also the time domain analysis from
numerical relativity, we identify the hydrodynamic normal modes and calculate
out the sound speed, which is shown to increase with the chemical potential and
saturate to the value predicted by the conformal field theory in the large
chemical potential limit. Third, the generic non-thermalization is demonstrated
by the fully non-linear time evolution from a non-equilibrium state for our
zero temperature holographic superfluid. Furthermore, a conserved Noether
charge is proposed in support of this behavior.
