A stone drops into a pond — ripples spread out and vanish. But in a narrow channel, a wave sometimes gathers into a hump and rolls on, unchanging in shape, kilometer after kilometer. Such stubborn solitary waves are called solitons. And if the crest is not smooth but pointed, like a spring compressed to its limit: that's a peakon.

In real life, channels are not perfect: somewhere shallower, somewhere the current is faster. The mathematicians took the Camassa–Holm equation with variable coefficients (vcCH) and added weak dispersion — a property of the medium that makes waves of different lengths travel at different speeds. Usually this smears the wave into a mush, but here small dispersion allowed them to seek solutions resembling solitons and peakons in the form of neat corrections. The main trick is to isolate the "singular" part responsible for the sharpness of the peak or the hump of the soliton, and separate it from the smooth background common to all such waves.

The authors constructed asymptotic series, as if assembling the wave from parts of ever finer adjustment, and proved how accurate these approximations are. For a single wave, they managed to write out corrections of any order and confirm that they make sense in appropriate mathematical spaces. For two interacting waves, a path is also outlined. At the end of the article are vivid examples with explicit formulas and graphs: the waves indeed hold their shape despite the changing medium.