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Up and out early from a long winter hibernation, Barney the Bear
has a Grand Adventure making many new animal friends, but must
return home before Momma and Poppa Bear awake.
The soliton is a dramatic concept in nonlinear science. What makes
this book unique in the treatment of this subject is its focus on
the properties that make the soliton physically ubiquitous and the
soliton equation mathematically miraculous. Here, on the classical
level, is the entity field theorists have been postulating for
years: a local traveling wave pulse; a lump-like coherent
structure; the solution of a field equation with remarkable
stability and particle-like properties. It is a fundamental mode of
propagation in gravity-driven surface and internal waves; in
atmospheric waves; in ion acoustic and Langmuir waves in plasmas;
in some laser waves in nonlinear media; and in many biologic
contexts, such as alpha-helix proteins. This is not an encyclopedia
of information on solitons in which every sentence is interrupted
by either a caveat or a reference. Rather, Newell has tried to tell
the story of the soliton as he would have liked to have heard it as
a graduate student, with some historical development, lots of
motivation, and frequent attempts to relate the topic at hand to
the big picture. The book begins with a history of the soliton from
its first sighting to the discovery of the inverse scattering
method and recent ideas on the algebraic structure of soliton
equations. Chapter 2 focuses on the universal nature of these
equations and how and why they arise in physical and engineering
contexts as asymptotic solvability conditions. The third chapter
deals with the inverse scattering method and perturbation theories.
Chapter 4 introduces the t-function and discusses the relations
between the various methods for constructing solutions to the
soliton equations and their various properties. Finally, an
algebraic structure for the equations is provided in Chapter 5.
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