The best evidence I have found that they may exist is . . .
“On a class of solutions of the sine-Gordon equation” Mikhail Kovalyov Journal of Physics A: Mathematical and Theoretical Volume 42 Number 492009 J. Phys. A: Math. Theor. 42 495207 doi:10.1088/1751-8113/42/49/495207
This article talks about anti-breathers. It basically says that there are real and imaginary breathers and anti-breathers in the what I have been calling the Super-breather. There are only two articles that I could find that spoke of anti-breathers at all. But it took me days to figure out what search term to use to find this article. But having just one article that claims that they exist is hardly sufficient.
Why is it a good question whether Superbreathers exist? I have had a theory since around 1995 that predicted that Superbreathers existed. And every now and again I try to look for someone who has discovered them. But recently I was writing an article where I suddenly needed to know whether they existed for my argument to hold weight. So I started a frantic search. It turns out in the meantime a lot has been learned about solitons and breathers, and all sorts of odd types of solitons and breathers have been discovered, but no mention of the missing predicted superbreather. In my formulation a super-breather is a combination of a breather and an anti-breather just as a breather is a combination of a soliton and an anti-soliton. These are mathematical and physical anomalies which are waves that are both particle and wave at the same time in macro phenomena.
Why do we care? Because like super-conductivity, solitons, breathers, and if they exist super-breathers call into question some of our traditional views of science. Like super-conductivity or Bose-Einstein Condensates they call into question our ideas about Thermodynamics. They are counterexamples and as such they are important. It took twenty years to come up with Cooper Pairs as the explanation of superconductivity, and it took us a very long time to realize that solitons are everywhere in physical equations, and to work out how to tease them out and then find out whether the equations described something real or not once the math was known. Now we know pretty much that the equations are correct and solitons and higher order derivatives like breathers and other strange soltonic phenomena are just about everywhere in physics. They are important because they are unexpected side effects of our physical equations that are counter intuitive. When we tease them out and find out that they are real that means our equations are better than we thought they were at predicting reality. This is a very fortuitous and unexpected finding that says that our equations really do predict the nature of the world, even in ways we did not intend in the first place when we created those equations.