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:
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.