  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
$\mathbb R$ 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 flow and the corresponding Airy flow is 1-smoothing.
