Modified scattering and beating effect for coupled Schrödinger systems on product spaces with small initial data

Abstract

In this paper, we study a coupled nonlinear Schrödinger 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 Grébert, Paturel, and Thomann for the study of the resonant system.