Modified scattering and beating effect for coupled Schr\"odinger systems on product spaces with small initial data

TL;DR

This paper investigates the long-term behavior of solutions to a coupled nonlinear Schr"odinger system on product spaces with small initial data, establishing modified scattering, constructing wave operators, and revealing dynamics like the beating effect.

Contribution

It introduces a new analysis of coupled Schr"odinger systems on product spaces, demonstrating modified scattering and detailed asymptotic dynamics including the beating effect.

Findings

Established modified scattering for the system

Constructed a modified wave operator

Identified the beating effect in the dynamics

Abstract

In this paper, we study a coupled nonlinear Schr\"odinger system with small initial data in a product space. We establish a modified scattering of the solutions of this system and we construct a modified wave operator. The study of the resonant system, which provides the asymptotic dynamics, allows us to highlight a control of the Sobolev norms and interesting dynamics with the beating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov and Visciglia for the modified scattering, and a recent work of Gr\'ebert, Paturel and Thomann for the study of the resonant system.