Does the VL System on submarines work the same way, like on ships.
Hot Launch:
The Missile turbo-fan starts burning inside the VLS Tubes, right?
But what happend with the hot exhaust gas?
And which function have the canister, is it used by hot or cold launch?
Hot launch systems are currently only used for missiles on surface combatants. The problem as you noted is what happens to the exhaust gases, which have to be vented separately from the missile.
The new AUC designs solves the exhaust gas problem, so we may finally see hot launch systems on sub. AUC is especially useful for small(er) missiles like SAMs.
Missiles with air breathing propulsion (prop, turbo fan, turbine engine, ramjet, scramjet, etc) use a rocket booster to launch and get up to flight speed quickly. The air breathing engine then deploys and takes over.
For sub launched missiles the canister is strictly for handling protection and as a moisture seal.
Cold Launch:
How I eject the missile. I mean the missile have to cross the way between sub and surface and than it have to breach through the water surface area! Therefore we need a lot of power, or not.
But I think this can be solved with compressed air, the air volume expanded on the way to surface.
Toward The Principe of Archimedes the missile have to getting faster and faster til it reach the surface.
2 systems: un-encapsulated and encapsulated.
Un-encapsulated systems are commonly used for SLBM. The missile silo is pressured using either compressed air or a gas generator until the pressure inside exceeds the pressure outside by a predetermined amount. The inner seal on the silo, which is basically a rupture disk, then releases the pressure ahead of the missile which them functions as a piston in a cylinder and is propelled outward by the gas pressure behind it (there should be a void area below the missile to function as an accumulator to aid this). The missile rises until the tip breaches the surface then the booster ignites.
Encapsulated systems are mostly for small missiles like Harpoon and Tomahawk. The capsule is a waterproof enclosure that rises rapidly to the surface at a predetermined angle propelled by buoyancy. When the capsule breaches the surface the booster fires propelling it into the air, where other charges shred the capsule releasing the missile. Many of these systems are designed to launch out of torpedo tubes, which is partially responsible for the growth in the size of them in submarines.