This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu 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,

Penn. (1985)

M.J. Ablowitz and P.A.Clarkson, Solitons, nonlinear evolution equations and

inverse

scattering, Cambridge (1991).

http://www.cup.org/titles/catalogue.asp?isbn=0521387302

See http://www.ma.hw.ac.uk/solitons/

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