EXPLICIT SOLUTIONS OF THE KORTEWEG-DE VRIES EQUATION WITHOUT SCATTERING DATA

We consider the problem of constructing solutions of the Korteweg-de Vries equation with the help of the inverse-scattering theory for the Schrödinger equation on the line. Using a scheme proposed by Zakharov and Shabat we show how solutions of the full equation are related to solutions of the linearized equation. In this way we obtain solutions which are not recovered by the standard IST method. Explicit solutions, other than the multisolitons are derived without referring to a set of scattering data: they decrease exponentially at both ends of the line, at the cost of a singularity whose location varies with time. © 1979 Società Italiana di Fisica.