If I take a bottle of carbonated water and open the cap slightly to allow the gas to escape, there will be a sudden rush of bubbles foaming to the top of the liquid for a few seconds, then a partial lull for a few seconds, then another smaller rush of bubbles, then a smaller lull, etc, like a damped oscillation. What causes this pulsation?
The model used in the paper is that for a bubble to form it must nucleate then grow to a size big enough to rise to the surface. This process takes a finite time. When the cap is loosened the decrease in pressure causes a burst of nucleation, but after a few seconds the bubbles bursting on the surface raise the pressure again and this inhibits nucleation. As the gas escapes through the loosened cap the pressure falls again and nucleation restarts. The process repeats until the amount of gas being released by the bubbles is not longer enough to inhibit nucleation.
Note that the paper doesn't go into the physics of the nucleation and growth process. It just assumes that this is the basis of the oscillation. However the assumption seems reasonable. Suppose you take a bottle of soda and momentarily loosen then tighten the caps. You'll see a burst of fizzing. Loosen then tighten the cap again and you'll get another burst, and so on. In this case you're causing the pressure to oscillate by manually controlling the gas flow, but you can see how you'd get a similar effect if the gas flow though a slightly loosened cap was just right.
It would be interesting to measure the pressure above a saturated CO$_2$ solution as a function of gas flow through the cap. I don't have the kit to do this, but it wouldn't be a difficult experiment.