On the low-regularity global well-posedness of a system of nonlinear Schrodinger Equation

In this article, we study the low-regularity Cauchy problem of a one dimensional quadratic Schrodinger system with coupled parameter $\alpha\in (0, 1)$. When $\frac{1}{2}<\alpha<1$,we prove the global well-posedness in $H^s(\mathbb{R})$ with $s>-\frac{1}{4}$, while for $0<\alpha<\frac{1}{2}$, we obtain global well-posedness in $H^s(\mathbb{R})$ with $s>-\frac{5}{8}$. We have adapted the linear-nonlinear decomposition and resonance decomposition technique in different range of $\alpha$.