One smoothing property of the scattering map of the KdV on $\mathbb R$

TL;DR

This paper demonstrates that the scattering map for the KdV equation on the real line acts as a Fourier transform plus a smoothing operator in weighted Sobolev spaces, leading to a 1-smoothing property of the flow difference.

Contribution

It establishes the smoothing property of the KdV scattering map and its implications for the flow difference, extending understanding of the equation's regularity features.

Findings

Scattering map is a perturbation of Fourier transform by a regularizing operator.

Difference between KdV flow and Airy flow is 1-smoothing.

Results hold in weighted Sobolev spaces without bound states.

Abstract

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.