In both Dead Man's Chest and At World's End, the Flying Dutchman is shown to use chaser cannons in the bow (in the films, they're referred to as "bow cannons" or "triple guns"). What I find particularly interesting about them is that they're essentially three barrels which rotate and fire sequentially. Was such a design ever invented or used by any navy in the age of sail?
As an overarching question, how did ships in the Age of Sail attempt to engage an enemy sailing away from them? How successful were such pursuits, and what weapons assisted with this task?
For reference, here's a scene where said cannons are shown for a few seconds from the 1:46 mark.
I'm going to break this one down into to parts because it lets me talk more about the age of sail which I've always found interesting.
#Bow Chasers
Forward-firing cannons at the bow of the ship were actually in use though they were quite rare. Arguably the most famous ship from the age of sail - the HMS Victory - for example did not have such guns.
A good example of a ship that did have such weapons is the French ship the Soleil Royal. You can see in the reconstructed model at the Musée national de la Marine that the ship had bow chasers - with the bow chasers being very similar in arrangement to those seen in Pirates of the Caribbean. There are also paintings from the Louvre (I couldn't find an official site that shows them) that show stern chasers as well. Which is also something to keep in mind - the ship you are chasing may be firing back (the Soleil Royal had 2 bow chasers and 4 stern chasers).
So why didn't more ships have chase guns? Quite frankly, there's no good place to put them. The bow of the ship is, understandably, the narrowest section so you can only fit a limited number of guns. You can't place gun ports too low in the front or you will flood the ship. If you are using your bow-chasers you're chasing another ship meaning you have a much smaller target to hit (stern profile instead of broadside), with fewer weapons, that aren't very accurate. So their use was quite limited.
#Triple-barreled Cannons
edit: When I refer to #-pounder in terms of cannons, the # is the weight of the shot fired by said cannon, so a 32-pounder fires a 32 pound cannonball
These are pure fiction I'm afraid. Cannons were extremely heavy, powerful, and cumbersome. The triple barrel wouldn't really save you any time over a single cannon either. Looking at even the smallest cannons, a triple barrel would be exceedingly heavy. A 1-pounder brass cannon weighed 112 pounds (51kg) for example (so 336 pounds - 152kg - just for the barrels without the mount). But the cannons in the film look like cast iron and larger than a 1-pounder...so let's call it a 3 pounder, which officially weighed in at 812 pounds (368kg) or a whopping 2436 pounds (1105kg) for all three barrels. After the first 3 shots, you're looking at having to reload all 3 barrels which may be slightly faster than reloading after each shot 3 times. The cannons in the movie are reset to firing position extremely quickly (and, as you can see, those guns are heavy and had to be manually rolled forward). So you still have to reload 3 cannons and you still have to reset the cannon after each of the 3 shots - you might save seconds in an action that takes minutes per shot.
The tri-barrel also means these cannons would be as heavy as a 9-pounder but only capable of firing a 3 pound ball. And this is the real crux of the issue. The size of the gun was the key. To quote from British Smooth-Bore Artillery (which I used extensively to get the weights of the cannons above)
The ordnance committee of 1844 recommended that the 9-foot gun of 34 hundredweight be substituted occasionally for the 18-pounder in batteries where quick fire might be necessary against storming parties or boat attacks. The 8-1/2 foot, 33 cwt. gun, was recommended for use sometimes with the siege train since it was powerful enough to dismount artillery and required less ammunition (presumably meaning powder) than an 18- or 24-pounder gun. By 1857, except for some of the 12-pounders of 6feet, all of these guns were in storage, either in England or abroad.
They favored larger guns for ship-to-ship combat, even if that made them shorter range (edit: longer barrels were longer range so a larger cannon with a shorter barrel would be similar weight to the smaller cannon with a longer barrel). The history of naval warfare was a constant push to get the biggest guns possible on ships. The bigger the gun, the greater the force you are hitting the other ship/coastal artillery with.
The other problem with the tri-barreled setup is an engineering one. The rotating barrel introduces at least one extra moving part - which is a potential point of failure.
edit: I will say, the use of bow chasers on The Flying Dutchman in this film does make some sense considering the role it's supposed to play in the film. It's a ship for chasing down other ships (though it's a bit large for that in practical use).