Soliton theory and Hankel operators

Abstract

Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.