  We investigate the relaxation of holographic superfluids after quenches, when
the end state is either tuned to be exactly at the critical point, or very
close to it. By solving the bulk equations of motion numerically, we
demonstrate that in the former case the system exhibits a power law falloff as
well as an emergent discrete scale invariance. The later case is in the regime
dominated by critical slowing down, and we show that there is an intermediate
time-range before the onset of late time exponential falloff, where the system
behaves similarly to the critical point with its power law falloff. We further
postulate a phenomenological Gross-Pitaevskii-like equation that is able to
make quantitative predictions for the behavior of the holographic superfluid
after near-critical quenches. Intriguingly, all parameters of our
phenomenological equation which describes the non-linear time evolution may be
fixed with information from the static equilibrium solutions and linear
response theory.
