Wave without trough? Why does this video appear to show a wave with no trough? Do such waves exist?

 A: Definitely what you can see there exists. It is an image of a real experiment on solitons in fluids. I think that most people will call it a wave. However, the equation satisfied by such a wave or soliton is not the ordinary wave equation, but a different differential equation. So if you call it or not a wave, it is a matter of convention, but its existence is a fact.
If you define something as a "wave" by saying that it satisfies the wave equation the problem is moved to "what do you call a wave equation" as there is no unique equation satisfied by all the "waves". I would not worry to much about names. The question of "is this really a wave" does not bring much benefit.
But you recognized that this is a peculiar type of wave and the first people to observe it were also thrilled to see it. It was first described at the beginning of the 19th century. Not that they did not exist before that time.
There is at least on thread on this stackexchange about the general question "what do you call a wave"?
A: A wave has both crest and trough (undulating form) portions of the same amplitude and frequency.
A wave with zero amplitude and zero frequency i.e. a monochromatic infinite straight line (if we are allowed to call it a wave!) will have no trough and compulsorily no crest.
