Computer Algebra Recipes

Chapter 7
The Hunt for Solitons
There is no better ... door by which you can enter into the study of
natural philosophy than by considering the ... physical phenomena of
acandle.
Michael Faraday, English physicist (1791{1867)
Nonlinear PDEs display a rich spectrum of solutions that in most cases must be
obtained by numerical means. However, there exist special analytic solutions
to some nonlinear PDEs of physical interest, the best known being soliton solutions
of nonlinear wave equations. A soliton is a stable solitary wave, which
is a localized pulse solution that can propagate at some characteristic velocity
without changing shape despite the \tug of war" between \competing terms"
in the governing equation of motion.
A simple physical example [EJMR81] of a solitary wave is provided by the
°ame of an ordinary lit candle. There exists a dynamic balance between the
di®usion of the heat from the °ame into the wax and the nonlinear energy
release as the wax vaporizes. The candle °ame advances into the wax at a
velocity that just maintains the balance. To check whether a solitary wave
is stable, i.e., is a soliton, one can subject the solitary wave to some type of
perturbation and see whether its integrity is preserved. For example, the candle
°ame may °icker because of an ambient air current, but it tends to preserve its
shape as the candle burns, so the °ame displays soliton-like behavior.
Of course, there exist many di®erent possible stability criteria that could be
invoked to decide whether a solitary wave is a soliton. Historically, however,
mathematicians have decided that in order for a solitary wave to be deemed
worthy of the name soliton, it must survive a collision with another solitary-wave
solution of the same PDE completely unchanged in shape. There are two main
approaches to applying this collisional stability criterion, either numerically or
analytically. The numerical simulation approach will be brie°y illustrated in the
last section. Analytic methods are considerably more complicated to implement
(see, e.g., [EM00]) and we will be content here only to quote some of the results
in the form of two-soliton solutions.
Given a nonlinear PDE, how do we know that it even has the possibility
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