Not that I know of. This mechanism uses boosters in the nose to "pull" the missile out of the water and gain some altitude. Historically, SLBM launches have been something like:
1. The submarine would surface, open hatches and launch missiles like a road mobile TEL.
2. The submerged-submarine would eject the missile out of the tube, and the missile's first stage would ignite while still in water.
3. Modern SLBMs are ejected by high-pressure gas-generating mechanisms out and above water (from upto 150ft depth). Once the missile clears water and stabilizes, the first stage is ignited.
EDIT: So I searched for that mechanism, it is called a cavitator (basically an explosive gas generator, which pushes the water aside to facilitate free upward motion of the missile, and is ejected after the missile's ejection). It is particularly associated with Russian SLBMs (R-39 Sineva).
However after comparing the Russian missile launches with this one, it seems that K-4's cavitator has much much more thrust, as it seems to physically propel the missile upward. Probably the design is a new or experimental one.