1981/05/01 by John W. Miles · 201 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Advanced Mathematical Physics Problems #Korteweg–de Vries equation #Physics #Inverse scattering transform #Dispersion relation #Mathematical physics #Dispersionless equation #Inverse scattering problem #Soliton #Nonlinear system #Inverse #Dispersion (optics) #Wave packet #Kadomtsev–Petviashvili equation #Classical mechanics #Mathematical analysis #Scattering #Quantum mechanics #Burgers' equation #Mathematics
paper · doi:10.1017/s0022112081001559
published in Journal of Fluid Mechanics 106, 131-147 (Cambridge University Press)
openalex publication_date 1981/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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 = c 0 (1 − β k 2 ) 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.