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In this paper, we study a coupled nonlinear Schr\"odinger system with smallinitial data in a product space. We establish a modified scattering of thesolutions of this system and we construct a modified wave operator. The studyof the resonant system, which provides the asymptotic dynamics, allows us tohighlight a control of the Sobolev norms and interesting dynamics with thebeating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov andVisciglia for the modified scattering, and a recent work of Gr\'ebert, Patureland Thomann for the study of the resonant system.

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