Cross-invariant manifolds and instability of the standing waves for a coupled Schrödinger system

This paper studies the coupled Schrödinger system$ \begin{equation*} \left\{ \begin{split} &-i\varphi_{t} = \Delta\varphi+\mu_{1}|\varphi|^{2}\varphi+\beta|\psi|^{2}\varphi, \quad (t, x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, \\ &-i\psi_{t} = \Delta\psi+\mu_{2}|\psi|^{2}\psi+\beta|\varphi|^{2}\psi, \quad (t, x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, \end{split} \right. \end{equation*} $which appears in binary mixtures of Bose-Einstein condensates or in nonlinear optics. By establishing cross-invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solutions is derived. Then, the instability of the ground states in the system is shown.