The nonlinear Schrödinger equation with a potential

Pierre Germain, Fabio Pusateri, Frédéric Rousset

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
DOIs
StateAccepted/In press - Jan 1 2018

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Nonlinear equations
Nonlinear Equations
Long-time Asymptotics
Small Solutions
Fourier transforms
Stationary Phase
Cubic equation
Spectral Measure
Scattering Matrix
Scattering
Long-time Behavior
Bound States
Fourier transform
Asymptotic Behavior
Infinity
Estimate

Keywords

  • Distorted Fourier transform
  • Modified scattering
  • Nonlinear Schrödinger equation
  • Scattering theory

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics

Cite this

The nonlinear Schrödinger equation with a potential. / Germain, Pierre; Pusateri, Fabio; Rousset, Frédéric.

In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 01.01.2018.

Research output: Contribution to journalArticle

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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.

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