The nonlinear Schrödinger equation with a potential

Germain, Pierre; Pusateri, Fabio; Rousset, Frédéric

doi:10.1016/j.anihpc.2017.12.002

Germain, Pierre ; Pusateri, Fabio ; Rousset, Frédéric

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1477-1530.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2017.12.002

Classification : 35Q55, 35B34
Mots clés : Nonlinear Schrödinger equation, Distorted Fourier transform, Scattering theory, Modified scattering

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     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},
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2017.12.002/

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