TY - JOUR

T1 - The nonlinear Schrödinger equation with a potential

AU - Germain, Pierre

AU - Pusateri, Fabio

AU - Rousset, Frédéric

N1 - Funding Information:
Acknowledgments. We thank Z. Hani for communicating to us that Cuccagna, Georgiev and Visciglia had announced a result for the case of odd solutions and even potentials [9]. We thank A. Stefanov for letting us know about the paper [40]. P.G. was partially supported by the NSF grant DMS-1501019. F.P. was partially supported by the NSF grant DMS-1265875.

PY - 2018/9

Y1 - 2018/9

N2 - We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.

AB - We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.

KW - Distorted Fourier transform

KW - Modified scattering

KW - Nonlinear Schrödinger equation

KW - Scattering theory

UR - http://www.scopus.com/inward/record.url?scp=85041592020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041592020&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2017.12.002

DO - 10.1016/j.anihpc.2017.12.002

M3 - Article

AN - SCOPUS:85041592020

VL - 35

SP - 1477

EP - 1530

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 6

ER -