### 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 language | English (US) |
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Pages (from-to) | 1477-1530 |

Number of pages | 54 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 35 |

Issue number | 6 |

DOIs | |

State | Published - Sep 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Applied Mathematics

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## Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*35*(6), 1477-1530. https://doi.org/10.1016/j.anihpc.2017.12.002