Computer Algebra Recipes

300 CHAPTER 7. THE HUNT FOR SOLITONS
equation models is much more di±cult, and analytic approaches have consisted
mainly in ¯nding special solutions, some of which are of physical importance.
Many of these approaches are quite complicated in nature, but the seeking of
analytic solitary-wave pro¯les is relatively easy, provided that analytic solutions
exist. This will now be illustrated in the next two examples.
7.2.1 Follow That Wave!
It's just a job ... waves pound the sand. I beat people up.
Muhammad Ali, American boxer, New York Times, 6April1977
Probably the ¯rst reported observation of soliton behavior recorded in the scienti¯c
literature was made by the Scottish engineer and naval architect John
Scott Russell [Rus44]. In the less formal style of scienti¯c reporting of the day,
he wrote:
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-de¯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 ¯gure 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 ¯rst chance interview with that
singular and beautiful phenomenon ::::
The \narrow channel" referred to by Russell still exists, being the Union Canal
linking Edinburgh with Glasgow. Actually, Russell was not observing the
\rapidly drawn boat" by accident, but was actually carrying out a series of experiments
to determine the force{velocity characteristics of di®erently shaped
boat hulls in order to determine design parameters for conversion from horse
power to steam power. His solitary-wave observations were followed by extensive
wave-tank experiments in which he established the major properties of
hydrodynamic solitary waves. [EJMR81]
The detailed mathematical explanation of Russell's solitary wave had to wait
50 years until 1895, when the relevant nonlinear Korteweg{de Vries equation,
@Ã @Ã
+ ®Ã
@t @x + @3Ã =0; (7.2)
@x3 was derived by the Dutch mathematicians Diederik Korteweg and Gustav de
Vries. In the KdV equation, Ã is the transverse displacement of the horizontal
water surface, x the spatial coordinate in the direction of wave propagation, t