Korteweg and de Vries proved Russell was correct by finding explicit, closed-form, travelling-wave solutions to their equation that moreover decay rapidly and so represent a highly localized moving hump.
Both the fact that such a solution to a non-linear equation could exist and the fact that one could write it explicitly were later to be recognized as extremely important, but they went relatively unnoticed at the time.
The KdV equation did not receive significant futher attention until 1965, when N. Zabusky and M. Kruskal published the results of their numerical experimentation with the equation. Their computer generated approximate solutions to the KdV equation indicated that any localized initial profile that was allowed to evolve according to the KdV equation eventually consisted of a finite set of localized travelling waves of the same shape as the original solitary waves discovered in 1895. Furthermore, when two of the localized disturbances collided, they would emerge from the collision again as another pair of travelling waves with a shift in phase as the only consequence of their interaction. Since the "solitary waves" which made up these solutions seemed to behave like particles, Zabusky and Kruskal coined the name "soliton" to describe them.
Shortly after that, another remarkable discovery was made concerning the KdV equation. A paper by C. Gardner, J. Greene, M. Kruskal, and R. Miura demonstrated that it was possible to write many exact solutions to the equation by using ideas from scattering theory. In particular, the solutions shown on these pages are exact solutions that can be found by this method. In modern terminology, we would say that they discovered the first integrable non-linear partial differential equation.
Since then many other equations have been found to be integrable and admit soliton solutions. However, the KdV equation is considered the canonical example, in part because it was the first equation known to have these properties.
Note: Although the description given above is the "standard" version of the story, I have come to doubt one piece of it. It has now been recognized that French mathematician Boussinesq had demonstrated the existence of solitary wave solutions to the "KdV equation" many years prior to the paper by Korteweg and de Vries. Perhaps Korteweg and de Vries were unaware of this research and were still attempting to answer the question of whether such solutions to nonlinear equations could exist, in which case the description above is still entirely correct (even though the equation is seemingly misnamed). However, I find it more likely that they knew Boussinesq's research well (they do cite it!) and were instead trying to answer a different question: How are the periodic waves and the solitary waves related? Their paper does answer that question in terms of elliptic functions.
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