Now Mars is pulsing 3 times per night in the UV, can anyone explain these waves in simple terms? Do these waves happen on Earth?
The Martian nightglow comes from molecules in the atmosphere that react with solar radiation, carry it over to the night side, and give off the energy they gained as ultraviolet light as they free-form stable compounds.  From https://www.nasa.gov/press-release/goddard/2020/mars-nightglow:


The diagram explains the cause of Mars’ glowing nightside atmosphere. On Mars’ dayside, molecules are torn apart by energetic solar photons. Global circulation patterns carry the atomic fragments to the nightside, where downward winds increase the reaction rate for the atoms to reform molecules. The downwards winds occur near the poles at some seasons and in the equatorial regions at others. The new molecules hold extra energy which they emit as ultraviolet light.
Credits: NASA/MAVEN/Goddard Space Flight Center/CU/LASP

The Martian nightglow has been used to trace atmospheric circulation on Mars.
As for a similar phenomenon on Earth, and for that matter on planets generally, the above NASA reference goes on to say:

Many planets including Earth have nightglow, but MAVEN is the first mission to collect so many images of another planet’s nightglow.