Solitons are solutions of a special class of nonlinear partial differential
equations (soliton equations, the best example is the KdV equation). They
are waves but behave like particles. The term “soliton” combining the beginning
of the word “solitary” with ending “on” means a concept of a fundamental
particle like “proton” or “electron”.
The events: (1) sighting, by chance, of a great wave of translation,
“solitary wave”, in 1834 by Scott–Russell, (2) derivation of KdV equation by
Korteweg de Vries in 1895, (3) observation of a very special type of wave interactions
in numerical experiments by Kruskal and Zabusky in 1965, (4) development of the
inverse scattering method for solving initial value problems by Gardener, Greene, Kruskal
and Miura in 1967, (5) formulation of a general theory in 1968 by P.D. Lax and
(6) contributions to deep theories starting from the work by R. Hirota (1971-74)
and David Mumford (1978-79), which also gave simple methods of solutions of
soliton equations, led to the development of one of most important areas of mathematics in
the 20th century.
This also led to a valuable application of solitons to physics, engineering and technology.
There are two aspects of soliton theory arising out of the KdV Equation:
- Applied mathematics – analysis of nonlinear PDE leading to dynamics of waves.
- Pure mathematics – algebraic geometry.
It is surprising that each one of these can inform us of the other in the intersection
that is soliton theory, an outcome of the KdV equation.
The subject is too big but I shall try to give some glimpses (1) of the history,
(2) of the inverse scattering method, and (3) show that an algorithm based on
algebraic-geometric approach is much easier to derive soliton solutions.