Mathematical Research Letters

Volume 23 (2016)

Number 3

Modified scattering for the cubic Schrödinger equation on product spaces: the nonresonant case

Pages: 841 – 861

DOI: https://dx.doi.org/10.4310/MRL.2016.v23.n3.a13

Authors

Benoît Grébert (Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, France)

Éric Paturel (Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, France)

Laurent Thomann (Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, France)

Abstract

We consider the cubic nonlinear Schrödinger equation on the spatial domain $\mathbb{R} \times \mathbb{T}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani–Pausader–Tzvetkov–Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant.

Keywords

modified scattering, nonlinear Schrödinger equation, small divisors, normal form

2010 Mathematics Subject Classification

35B40, 35Q55

Full Text (PDF format)

Accepted 9 June 2015

Published 8 July 2016