This article is from the Nonlinear Science FAQ, by James D. Meiss email@example.com with numerous contributions by others.
The process of obtaining a solution of a linear (constant coefficient)
differential equations is simplified by the Fourier transform (it converts
such an equation to an algebraic equation, and we all know that algebra is
easier than calculus!); is there a counterpart which similarly simplifies
nonlinear equations? The answer is No. Nonlinear equations are qualitatively
more complex than linear equations, and a procedure which gives the dynamics
as simply as for linear equations must contain a mistake. There are, however,
exceptions to any rule.
Certain nonlinear differential equations can be fully solved by, e.g., the
"inverse scattering method." Examples are the Korteweg-de Vries, nonlinear
Schrodinger, and sine-Gordon equations. In these cases the real space maps, in
a rather abstract way, to an inverse space, which is comprised of continuous
and discrete parts and evolves linearly in time. The continuous part typically
corresponds to radiation and the discrete parts to stable solitary waves, i.e.
pulses, which are called solitons. The linear evolution of the inverse space
means that solitons will emerge virtually unaffected from interactions with
anything, giving them great stability.
More broadly, there is a wide variety of systems which support stable solitary
waves through a balance of dispersion and nonlinearity. Though these systems
may not be integrable as above, in many cases they are close to systems which
are, and the solitary waves may share many of the stability properties of true
solitons, especially that of surviving interactions with other solitary waves
(mostly) unscathed. It is widely accepted to call these solitary waves
solitons, albeit with qualifications.
Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon.
Many very basic linear systems with the addition of the simplest possible or
first order nonlinearity support solitons; this universality means that
solitons will arise in many important physical situations. Optical fibers can
support solitons, which because of their great stability are an ideal medium
for transmitting information. In a few years long distance telephone
communications will likely be carried via solitons.
The soliton literature is by now vast. Two books which contain clear
discussions of solitons as well as references to original papers are
A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia,
M.J. Ablowitz and P.A.Clarkson, Solitons, nonlinear evolution equations and
scattering, Cambridge (1991).