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.
Publisher Copyright:
© 2018 Elsevier Masson SAS
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
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U2 - 10.1016/j.anihpc.2017.12.002
DO - 10.1016/j.anihpc.2017.12.002
M3 - Article
AN - SCOPUS:85041592020
SN - 0294-1449
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
IS - 6
ER -