Contents Online
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
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
Accepted 9 June 2015
Published 8 July 2016