The purpose of this talk is to propose a study of various nonlinear behav-

iors for a system of two coupled cubic Schr ̈odinger equations with small initial data.

Depending on the choice of the spatial domain, we highlight different examples of non-

linear behaviors. On the one hand, we observe on the torus a truly nonlinear behavior

(exchanges on energy) in finite time. On the other hand, on the real line, we highlight

through scattering methods an almost linear behavior in infinite time. The goal is to

mix these two approaches to obtain on the product space a truly nonlinear behavior in

infinite time, via the construction of a modified scattering theorem.