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

Alberto  Maspero; Beat  Schaad

doi:10.3934/dcds.2016.36.1493

Article Contents

2016, Volume 36, Issue 3: 1493-1537. Doi: 10.3934/dcds.2016.36.1493

This issue Previous Article Young towers for product systems Next Article On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid

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

Alberto Maspero^1, and
Beat Schaad^2,

1.
Department of Mathematics, Università la Sapienza Roma, Piazzale Aldo Moro, 5, 00185 Rome
2.
Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7594

Received: December 2014

Revised: June 2015

Published: May 2017

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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.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35P25, 37K15.

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