One smoothing property of the scattering map of the KDV on R

In this paper we prove that in appropriate weighted Sobolev spaces, in the case of no bound states, the scattering map of the Korteweg-de Vries (KdV) on R is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV ow and the corresponding Airy ow is 1-smoothing.