Global well-posedness and scattering for the coupled NLS in critical spaces

In this paper, we mainly concern with the system of the coupled nonlinear Schrödinger equation with initial data in critical spaces in dimension 3. We first obtain the global well-posedness of the solution to the coupled nonlinear Schrödinger equation with initial data in a critical space W117,76(R3). The key is to derive a uniform bound of a modified energy E(t) based on a decomposition for the solution as in Dodson [Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. Rev Mat Iberoam. 2022;38:1087–1100]. In addition, inspired by Dodson [Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. International mathematics research notices; 2023], we make a new decomposition for the solution and show the scattering to the coupled nonlinear Schrödinger equation with radially symmetric initial data in a critical Besov space B1,12(R3).