Background: The discovery of solitary waves of translation goes back to Scott Russell in 1834, and during the remaining part of the 19th century the true nature of these waves remained controversial. It was only with the derivation by Korteweg and de Vries in 1895 of what is now called the Korteweg-de Vries (KdV) equation, that the one-soliton solution and hence the concept of solitary waves was put on a firm basis. 1 An extraordinary series of events took place around 1965 when Kruskal and Zabusky, while analyzing the numerical results of Fermi, Pasta, and Ulam on heat conductivity in solids, discovered that pulselike solitary wave solutions of the KdV equation, for which the name "solitons" was coined, interact elastically. This was followed by the 1967 discovery of Gardner, Greene, Kruskal, and Miura that the inverse scattering method allows one to solve initial value problems for the KdV equation with sufficiently fast-decaying initial data. Soon thereafter, in 1968, Lax found a new explanation of the isospectral nature of KdV solutions using the concept of Lax pairs and introduced a whole hierarchy of KdV equations. Subsequently, in the early 1970s, Zakharov and Shabat (ZS), and Ablowitz, Kaup, Newell, and Segur (AKNS) extended the inverse scattering method to a wide class of nonlinear partial differential equations of relevance in various scientific contexts ranging from nonlinear optics to condensed matter physics and elementary particle physics. In particular, solitons found numerous applications in classical and quantum field theory and in connection with optical communication devices. Another decisive step forward in the development of completely integrable soliton equations was taken around 1974. Prior to that period, inverse spectral 1 With hindsight, though, it is now clear that other researchers, such as Boussinesq, derived the KdV equation and its one-soliton solution prior to 1895, as described in the notes to Section 1.1.
