  For flowing quantum gases, it has been found that at long times an initial
black-hole laser (BHL) configuration exhibits only two possible states: the
ground state or a periodic self-oscillating state of continuous emission of
solitons. So far, all the works on this subject are based on a highly idealized
model, quite difficult to implement experimentally. Here we study the
instability spectrum and the time evolution of a recently proposed realistic
model of a BHL, thus providing a useful theoretical tool for the clear
identification of black-hole lasing in future experiments. We further confirm
the existence of a well-defined phase diagram at long times, which bespeaks
universality in the long-time behavior of a BHL. Additionally, we develop a
complementary model in which the same potential profile is applied to a
subsonic homogeneous flowing condensate that, despite not forming a BHL,
evolves towards the same phase diagram as the associated BHL model. This result
reveals an even stronger form of robustness in the long-time behavior with
respect to the transient, which goes beyond what has been described in the
previous literature.
