The Korteweg-de Vries equation: a historical essay

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Abstract
The Korteweg-de Vries (KdV) equation, usually attributed to Korteweg & de Vries (1895), governs the propagation of weakly dispersive, weakly nonlinear water waves and serves as a model equation for any physical system for which the dispersion relation for frequency vs. wavenumber is approximated by ω/k = c0(1 − βk2) and nonlinearity is weak and quadratic. It first appears explicitly in de Vries's dissertation (1894), although it is implicit in the work of Boussinesq (1872). Its current renaissance stems from the Fermi, Pasta & Ulam (1955) problem for a string of nonlinearly coupled oscillators, which, through the work of Zabusky & Kruskal and their colleagues, led to the discovery of the soliton and the development of inverse-scattering theory by Gardner et al. (1967). Many related evolution equations, each of which represents a balance between some form of dispersion (or variation of dispersion in the case of wave-packet evolution) and weak nonlinearity in an appropriate reference frame, have since been found to have properties analogous to those of the KdV equation - in particular, inverse-scattering solutions that are asymptotically dominated by solitons.