Active matter, which includes crowds of organisms, is composed of constituents that independently consume and dissipate energy. Some active matter systems have been shown to sustain the propagation of various types of waves, resulting from the interplay between density and alignment. Here, we examine a type of solitary wave in dense two-dimensional columns of Solenopsis invicta, fire ants, in which the local activity, density and alignment all play a key role. We demonstrate that these waves are nonlinear and that they are composed of aligned ants that are constrained at the top by the time it takes disordered ants to activate and align and at the bottom by a density minimum enforced by gravity. Our results suggest that intrinsically switchable activity can be a productive framework to understand and trigger a broad range of wave-like behaviors, including stampedes in crowds and herds.