Voir la notice de l'article provenant de la source Numdam
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.
Keywords: Nonlinear Schrödinger equation, Distorted Fourier transform, Scattering theory, Modified scattering
@article{AIHPC_2018__35_6_1477_0,
author = {Germain, Pierre and Pusateri, Fabio and Rousset, Fr\'ed\'eric},
title = {The nonlinear {Schr\"odinger} equation with a potential},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1477--1530},
publisher = {Elsevier},
volume = {35},
number = {6},
year = {2018},
doi = {10.1016/j.anihpc.2017.12.002},
mrnumber = {3846234},
zbl = {1406.35355},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.002/}
}
TY - JOUR AU - Germain, Pierre AU - Pusateri, Fabio AU - Rousset, Frédéric TI - The nonlinear Schrödinger equation with a potential JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1477 EP - 1530 VL - 35 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.002/ DO - 10.1016/j.anihpc.2017.12.002 LA - en ID - AIHPC_2018__35_6_1477_0 ER -
%0 Journal Article %A Germain, Pierre %A Pusateri, Fabio %A Rousset, Frédéric %T The nonlinear Schrödinger equation with a potential %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1477-1530 %V 35 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.002/ %R 10.1016/j.anihpc.2017.12.002 %G en %F AIHPC_2018__35_6_1477_0
Germain, Pierre; Pusateri, Fabio; Rousset, Frédéric. The nonlinear Schrödinger equation with a potential. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1477-1530. doi: 10.1016/j.anihpc.2017.12.002
Cité par Sources :