The field of nonlinear dispersive waves has developed rapidly over the past 50 years. Its roots go back to the work of Stokes in 1847, Boussinesq in the 1870s and Korteweg and de Vries (KdV) in 1895, all of whom studied water wave problems. In the early 1960s researchers developed effective asymptotic methods, such as the method of multiple scales, that allow one to obtain nonlinear wave equations such as the KdV equation and the nonlinear Schrödinger (NLS) equation, as leading-order asymptotic equations governing a broad class of physical phenomena. Indeed, we now know that both the KdV and NLS equations are “universal” models. It can be shown that KdV-type equations arise whenever we have weakly dispersive and weakly nonlinear systems as the governing system. On the other hand, NLS equations arise from quasi-monochromatic and weakly nonlinear systems.

The discovery of solitons associated with the KdV equation in 1965 by Zabusky and Kruskal was a major development. They employed a synergistic approach: computational methods and analytical insight. This was soon followed by a remarkable publication in 1967 by Gardner, Greene, Kruskal and Miura that described the analytical method of solution to the KdV equation, with rapidly decaying initial data. They employed concepts of direct and inverse scattering in the solution of the KdV equation that was perceived by researchers then as nothing short of astonishing.