The nonlinear Schrödinger equation with a potential

  • Pierre Germain

    Courant Institute of Mathematical Sciences, 251 Mercer Street, New York 10012-1185, NY, USA
  • Fabio Pusateri

    Department of Mathematics, Princeton University, Washington Road, Princeton 08540, NJ, USA
  • Frédéric Rousset

    Laboratoire de Mathématiques d'Orsay (UMR 8628), Université Paris-Sud, 91405 Orsay Cedex, France


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.

Cite this article

Pierre Germain, Fabio Pusateri, Frédéric Rousset, The nonlinear Schrödinger equation with a potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, pp. 1477–1530

DOI 10.1016/J.ANIHPC.2017.12.002