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.
as written by the authors arXiv:2604.17348
The paper deals with the Camassa--Holm equation with variable coefficients (vcCH equation) that is a direct generalization of the well known Camassa--Holm equation. We focus on the mathematical description of particular solutions of the vcCH equation with a small dispersion that exhibit properties analogous to those of classical soliton and peakon solutions, and consider the construction of soliton- and peakon-like solutions in the form of asymptotic expansions, including both one-phase and two-phase cases. The solution is expressed as the sum of a regular background common to all soliton- and peakon-like solutions and a singular component that captures their distinctive features, with the precise definition of the main singular term playing a central role. In the one-phase case, this term is determined, and the solvability of higher-order singular corrections is established in suitable functional spaces, enabling the construction of asymptotic solutions to arbitrary accuracy in a small parameter. The study also addresses the construction of two-phase soliton- and peakon-like solutions. Theorems on the asymptotic accuracy of the constructed asymptotic solutions have been proved. Each of the considered cases is illustrated by nontrivial examples for which, in accordance with the obtained general results, approximate solutions are derived in explicit form and their graphs are presented.
Solitons were first observed by Scottish engineer John Scott Russell in 1834: he saw a wave raised by a barge in a channel roll for several kilometers without changing shape, and he followed it on horseback.