Solitons: The KdV Equation

A soliton is a wave-packet that maintains its shape while propagating at a constant velocity. Broadly speaking, solitons have three defining features

•  They are of permanent form (i.e. static solutions of the field equations),

•  They are localised within a region (i.e. they have finite size and finite energy),

•  They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.

The KdV equation is a mathematical model of waves on shallow water surfaces and is notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. First written down and solved in the simplest case, by means of the inverse scattering transform, by Korteweg and de-Vires in 1895 to explain the following account of J. Scott Russell who observed a soliton while riding beside a narrow barge channel. The following passage has been taken from J . Scott Russell. Report on waves, Fourteenth meeting of the British Association for the Advancement of Science, 1844. "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-defined 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 which I have called the Wave of Translation".