Well-posedness of initial value problem of Hirota-Satsuma system in low regularity Sobolev space

In this paper, we study the initial value problem of Hirota-Satsuma system:

$$\{\begin{array}{ll}{u}_t-\alpha (u_{xxx}+6u{u}_x)=2\beta v{v}_x,& x\in \mathbb{R},t\ge 0,\\ v_t+v_{xxx}+3u{v}_x=0,& x\in \mathbb{R},t\ge 0,\\ u(0,x)=\varphi (x),v(0,x)=\psi (x),& x\in \mathbb{R},\end{array}$$

where $\alpha \in \mathbb{R}$, $\beta \in \mathbb{R}$; $u=u(x,t)$, $v=v(x,t)$ are real functions. Aided by Fourier restrict norm method, we show that $\mathrm{\forall }s>-\frac{1}{8}$ initial value problem (0.1) is locally well-posed in $H^s(\mathbb{R})\times H^{s+1}(\mathbb{R})$ which improved the results of ^[7].