Spectral Transform and Solitons: How to Solve and Investigate Nonlinear Evolution Equations

The soliton was discovered (and named) in 1965 by Zabusky and Kruskal,(1) who were experimenting with the numerical solution by computer of the Korteweg-de Vries (KdV) equation. This nonlinear partial differential equation had been introduced at the end of the last century to describe wave motion in shallow canals.(2) Zabusky and Kruskal studied the equation because of its relevance to plasma physics, as well as to the Fermi-Pasta-Ulam puzzle(3) (for a fascinating account of the motivations that led to the “birth of the soliton,” see Kruskal.(4)) The first scientific description of the soliton as a natural phenomenon, however, goes back to the first half of the nineteenth century, and was reported by J. Scott-Russell in the following prose:(5) I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon....