The basic operation of the PDE is similar to that of the pulsejet; air is mixed with fuel to create a flamable mixture that is then ignited. The resulting combustion greatly increases the pressure of the mixture, which then expands through a nozzle for thrust. To ensure that the mixture exits to the rear, thereby pushing the aircraft forward, the pulsejet uses a series of shutters or careful tuning of the inlet to force the air to travel only in one direction through the engine.
The main difference between a PDE and traditional pulsejet is the way in which the airflow and combustion in the engine is controlled. In the PDE the combustion process is supersonic, effectively an explosion instead of burning, and the shock wave of the combustion front inside the fuel serves the purpose of the shutters of a pulsejet. When the shock wave reaches the rear of the engine and exits the fuel is ejected in "one go", the pressure inside the engine suddenly drops, and air is pulled in the front of the engine to start the next cycle.
The main side effect of the change in cycle is that the PDE is considerably more efficient. In the pulsejet the combusion pushes a considerable amount of the fuel/air mix (the charge) out the rear of the engine before it has had a chance to burn (thus the trail of flame seen on the V1 missile), and even while inside the engine the mixture's volume is continually changing, an inefficient way to burn fuel. In contrast the PDE deliberately uses a high-speed combustion process that burns all of the charge while it is still inside the engine at a constant volume, a much more efficient process.
Another side effect, not yet demonstrated in practical use, is the cycle time. A traditional pulsejet tops out at about 250 pulses per second, but the aim of the PDE is thousands of pulses per second, so fast that it is basically continual from an engineering perspective. This should smooth out the otherwise highly vibrational pulsejet engine.
NASA maintains a research program on the PDE, which is aimed at high-speed, about mach 5, civilian transport systems. However most PDE research is military in nature, as the engine could be used to develop a new generation of high-speed, long-range reconnaissance aircraft that would fly high enough to be out of range of any current anti-aircraft defenses, while offering range considerably greater than the SR-71, which required a massive tanker support fleet to use in operation.
While most research is on the high speed regime, newer designs with much higher pulse rates in the hundreds of thousands appear to work well even at subsonic speeds. Whereas traditional engine designs always include tradeoffs that limit them to a "best speed" range, the PDE appears to outperform them at all speeds. Both Pratt and Whitney and General Electric now have active PDE research programs in an attempt to commercialize the designs.