Soliton

A soliton is a self-reinforcing solitary wave that maintains its shape while traveling at a constant velocity, arising from a precise balance between nonlinear effects and dispersion. Unlike ordinary waves, a soliton can interact with other waves and emerge from a collision without changing form, apart from a possible phase shift. The concept originated from observations of a long, solitary water wave by John Scott Russell in 1834 in a canal near Edinburgh. The term soliton was coined in 1965 by Zabusky and Kruskal, who observed soliton-like behavior in the Korteweg–de Vries equation during numerical experiments. Later, the development of the inverse scattering transform showed that many soliton equations are integrable and possess an infinite number of conserved quantities.

Solitons appear in a variety of nonlinear dispersive systems. Important examples include the Korteweg–de Vries equation, u_t + 6 u u_x + u_{xxx} = 0, and the nonlinear Schrödinger equation, i ψ_t + ψ_xx + 2|ψ|^2 ψ = 0. Depending on the system, solitons can be bright (localized peaks) or dark (localized dips), and there are topological solitons such as kinks in the sine-Gordon equation and other field theories.

Solitons have practical relevance in physics and engineering. They model long internal waves in shallow water, optical pulses in fibers, and matter-wave packets in Bose–Einstein condensates and plasmas. Their robustness and particle-like interactions make them useful both as a theoretical tool and as a practical technology in optical communications and nonlinear wave research.