Einstein made several predictions in his general theory of relativity, most of which were confirmed quite rapidly (the most famous one is considered to be the perihelion of Mercury, but the mere fact that GPS technology works should be enough!)
The one prediction that was not confirmed was the existence of gravitational waves. Despite the hundreds of millions spent to detect these funny invisible creatures, they just kept eluding us making their existence almost questionable (if they in fact do not exist, this will be bad for our friend, Dr. Einstein).
Recently, my advisor Vladimir Zakharov and myself published a paper proving the stability of a certain class of gravitational waves (Bondi-Robinson-Pirani). The proof reveals a striking similarity between this stability problem and the stability of the Scwarzschild black hole (proven by Regge and Wheeler in 1957). The most interesting conclusion, in my opinion, is that this work hints that this class of waves are not only stable but completely transparent to other gravitational waves. This means that under some conditions they may just go right through other gravitational waves! This property of transparency is a common property of solitons, and hints that perhaps Einstein's equations are integrable if the metric is diagonal (see the paper below for the full definition).