Solitons and the Korteweg-de Vries equation: Integrable systems in 1834--1995
This review of highlights from the history of the Korteweg-de Vries equation is entirely apt for a centennial. The history breaks into classical and modern periods with an intermediate stage, apparently unrelated at the time. The publication of the work of \textit{D. J. Korteweg} and \textit{G. de Vries} [Phil. Mag. 39, 422-443 (1895, JFM 26.0881.02)] actually brings the classical period to a conclusion. It started with the discovery in 1834 of the solitary water wave by \textit{J. S. Russell} [Trans. Royal Soc. Edinburgh, XIV, 47-109 (1840)] which started decades of controversy involving Russell, Stokes, Airy, Boussinesq, Lord Rayleigh and St. Venant. The authors present an engaging and relatively detailed discussion of the issues, which boil down to the mathematical description of a wave that may travel without distortion, resolved by the paper of 1895 and the explicit solutions therein. Between this phase and the modern lies a more abstract discussion concerning the ``integrability'' of nonlinear ordinary differential equations. This was due largely to Painlevé, Gambier and Fuchs and will be found in most older treatments of differential equations. The discovery that the KdV, under certain symmetry constraints, reduces to equations of such types gave them a new lease of life in the modern period. This starts with the work of \textit{N. J. Zabusky} and \textit{M. D. Kruskal} [Phys. Rev. Lett. 15 (1965)] on recurrence in nonlinear lattices and the almost immediate development by \textit{C. S. Gardner}, \textit{J. M. Green}, \textit{M. D. Kruskal} and \textit{R. M. Miura} [Phys. Rev. Lett. 19 (1967)] of the Inverse Scattering Method as a machine for solving the Korteweg-de Vries Cauchy problem. Following this singular event the subject of soliton mathematics and physics exploded. The authors therefore choose to discuss questions of particular concern to themselves notably, the applications of integrability to partition functions in statistical physics and integrable equations in higher dimensions (although the latter is sadly brief). The final section deals with the role of the KdV in 2-dimensional Quantum Gravity and conformal field theory.
scientific article; zbMATH DE number 790798
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