Modified scattering for the cubic Schr{\"o}dinger equation on product spaces: the nonresonant case

TL;DR

This paper proves modified scattering and constructs wave operators for the cubic nonlinear Schr{"o}dinger equation on mixed Euclidean-torus spaces with a convolution potential, showing that small solutions have asymptotically constant Sobolev norms.

Contribution

It extends modified scattering results to the nonresonant cubic NLS on product spaces with a convolution potential, using recent advanced techniques.

Findings

Sobolev norms of small solutions are asymptotically constant

Modified wave operators are constructed under generic potential assumptions

The analysis applies to the nonresonant case on rac{rac{R}{} imes rac{rac{T}^d}{} spaces.

Abstract

We consider the cubic nonlinear Schr{\"o}dinger 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.