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

Abstract

In this paper, we study the initial value problem of Hirota-Satsuma system: $ \begin{equation} \notag \left\{ \begin{array}{ll} u_t-\alpha(u_{xxx}+6uu_x) = 2\beta vv_x, & \ x\in {\mathbb{R}}, \ t\ge 0, \\ v_t+v_{xxx}+3uv_x = 0, & x\in {\mathbb{R}}, \ t\ge 0, \\ u(0, x) = \phi(x), \; \; v(0, x) = \psi(x), & x\in {\mathbb{R}}, \end{array} \right. \end{equation} $ 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 $ \forall s > -\frac 18 $ 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].

Keywords

• hirota-satsuma system
• initial value problem
• fourier restrict norm method
• local well-posed