Introduction

Abstract

In the late 60’s and 70’s a remarkable renaissance occurred around an equation, discovered in 1895 by Korteweg and de Vries, describing the evolution over time of a shallow water wave. This equation has its roots in Scott Russel’s horseback journey along the Edinburgh to Glasgow canal; he followed a wave created by the prow of a boat, which stubbornly refused to change its shape over miles. This revival in the 60’s was driven by a discovery of Kruskal and coworkers: the scattering data for the one-dimensional Schrödinger operator, with potential given by the solution of the KdV equation, moves in a remarkably simple way over time, while the spectrum is stubbornly preserved in time. This led to a Lax pair representation involving a fractional power of the Schrödinger operator; it ties in with later developments around coadjoint orbits in the algebra of pseudo-differential operators. Very soon it was realized that this isolated example of a “soliton equation” had many striking properties, leading to an explosion of ideas, following each other at a rapid pace.