$H^s_x\times H^s_x$ scattering theory for a weighted gradient system of 3D radial defocusing NLS

Abstract

In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering theories for the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t+\Delta u=\lambda |v|^2u,\quad iv_t+\Delta v=\mu|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3. \end{array}\right. \end{equation*} Here $\lambda>0$, $\mu>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.