solitons

A soliton is a self-reinforcing solitary wave that maintains its shape while traveling at constant speed due to a precise balance between nonlinear effects and dispersion. The phenomenon was first observed by John Scott Russell in a canal in 1834, and the term soliton was coined in the 1960s by Zabusky and Kruskal, following numerical experiments on the Korteweg–de Vries equation which showed solitary waves that survived collisions.

Many solitons arise in integrable nonlinear wave equations, such as the KdV equation, the nonlinear Schrödinger equation, and the sine-Gordon equation. These systems possess an infinite set of conserved quantities and can be solved by inverse scattering transform. Single-soliton solutions describe isolated waves; multi-soliton solutions describe several waves that interact elastically and emerge with only phase shifts. Solitons can be found in continuous media and in discrete lattices (discrete NLS, Frenkel-Kontorova).

Physical realizations include shallow-water waves, optical pulses in nonlinear fibers, plasma waves, and Bose–Einstein condensates. In optical communications, optical solitons enable dispersion-managed transmission over long distances. Not all solitary waves are solitons: in non-integrable systems, solitary waves can deform or dissipate after interactions. The study of solitons combines nonlinear physics, applied mathematics, and numerical methods, with ongoing research into novel materials and higher-dimensional and nonlocal solitons.