Solitons (or "solitary waves") are special solutions of certain nonlinear wave equations ( Korteweg-de-Vries, Benjamin-Ono, Kadomtsev-Petviashvili, the nonlinear Schroedinger, Sine-Gordon, etc., equations) that maintain their shape and amplitude as they propagate, because the solution maintains a balance between dissipation effects and wave dispersion, permanently ("standard" solitons), or on average ("breathers", etc.).
Discovered as water surface waves, in 1834 (Scott Russell 1844), they were much later theoretically explained by Boussinesq (in 1871-1877), Lord Rayleigh (in 1876), and by de Vries (in 1894; Korteweg and de Vries 1895). Then, they were nearly forgotten, until new interest in different fields arose, in the second half of the 20th century (Zabusky and Kruskal, 1965).
Today, the existence of solitons has been confirmed in the most diverse scientific contexts, and they play a considerable economic role by enabling very-high-speed communication via optical fibres which is part of the reason why you can read this, now.
Of course, a considerable literature dedicated to soliton theory exists, nowadays. This is not only about how to find analytical solutions of nonlinear differential equations, but also about the very surprising mathematics. To better understand the context and implications, I recommend a freely available book review by M[elvyn].S. Berger, especially for attempting to explain the slow dissemination of these very important advances, among the "general" scientific community (note: this review is more than thirty years old, but the "hypnotic effect" of "immensely successful linear (and linearizable) theories" -in Berger's words- still persists). The Wikipedia article on Martin David Kruskal is also good reading as an introduction to the mathematical importance of solitons.
The superposition of solitons is of special interest since 1965, but is also investigated recently (Benes et al. 2006; see also Andrei D. Polyanin's web, especially the text on multi-soliton solutions of the KdV equation), and the animations in Daniel Russell's web site.
A very insightful paper (Kshevetskii 2001) explains why solitons may tend to escape detection in a conventional numerical treatment of atmospheric waves. It is also unique in that many of the numerical results shown correspond to the mesopause region! The more recent joint paper by the same author, together with the expert on atmospheric gravity waves, Nikolai Gavrilov, shows how the same numerical treatment also includes, and is consistent with, traditional results obtained from linear wave theory.
N. Benes, A. Kasman, and K. Young 2006, On decompositions of the KdV 2-soliton, Journal of Nonlinear Science 16(2), 179-200.
M.S. Berger 1981, Bull.Am.Math.Soc. 4(3), 362-368.
J. Boussinesq 1871, Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Comptes Rendues 72, 755-759.
J. Boussinesq 1871, Comptes Rendues 73, 256-260.
J. Boussinesq 1872, J.Math.Pures et Appl.17, 55-108.
J. Boussinesq 1877, Essai sur la theorie des eaux courantes, Mem. Ac.Sci.Inst.Nat.France XXIII, 1-680.
E. de Jager 2006, On the origin of the Korteweg-de Vries equation, arX-iv:math.HO/0602661v1 28 Feb 2006.
G. de Vries 1894, Bijdrage tot de kennis der lange golven, 95 pp, PhD thesis, U. Amsterdam (publ. Loosjes, Haarlem).
D.J. Korteweg and G. de Vries 1895, On the change of form of long waves advancing in a rectangular canal, and on a new type of stationary waves, Philosophical Magazine 39, 422-443.
Kshevetskii, S.P., 2001, "Analytical and numerical investigation of nonlinear internal gravity waves", Nonlinear Processes in Geophysics 8, 37-53.
S.P. Kshevetskii and N.M. Gavrilov 2005, "Vertical propagation, breaking and effects of nonlinear gravity waves in the atmosphere", Journal of Atmospheric and Solar-Terrestrial Physics 67(11), 1014-1030.
Rayleigh (J.W. Strutt) 1876, On waves, Phil. Mag. 1, 257-271.
J. Scott Russell 1844, Report on waves, Rep. 14th meeting Brit. Assoc. Adv. Sci, J. Murray London, 311-390.
Zabusky, N.J., and M.D. Kruskal 1965, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Phys.Rev.Lett. 15, 240-243.