Asymptotic behavior of solutions to the cubic coupled Schr\"odinger systems in one space dimension

In this paper, we study a coupled nonlinear Schr{"o}dinger system with small initial data in the one dimension Euclidean space. Such a system appears in the context of the coupling between two different optical waveguides. We establish an asymptotic nonlinear behavior and a decay estimate for solutions of this system. The proof uses a recent work of Kato and Pusateri.