Local well-posedness of the coupled KdV-KdV systems on $ \mathbb{R} $

<p style='text-indent:20px;'>Inspired by the recent successful completion of study well-posedness theory for Cauchy problem Korteweg-de Vries (KdV) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +uu_x +u_{xxx} = 0, \quad \left. u \right |_{t 0} u_{0} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'> in space <inline-formula><tex-math id="M1">\begin{document}$ H^{s} (\mathbb{R}) $\end{document}</tex-math></inline-formula> (or id="M2">\begin{document}$ (\mathbb{T}) $\end{document}</tex-math></inline-formula>), we a class coupled KdV-KdV (cKdV) systems</p><p id="FE2"> \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, (u, v)\right (u_{0}, v_{0}) \end{array}\right. style='text-indent:20px;'>in id="M3">\begin{document}$ \mathcal{H}^s : H^s (\mathbb{R})\times $\end{document}</tex-math></inline-formula>. Typical examples include Gear-Grimshaw system, Hirota-Satsuma system and Majda-Biello to name few.</p><p In this paper look those values id="M4">\begin{document}$ s\in \mathbb{R} which cKdV systems are well-posed id="M5">\begin{document}$ ( \mathbb {R}) The key ingredients proofs bilinear estimates both divergence non-divergence forms under Fourier restriction norms. Sharp results established all four types that associated systems. contrast lone critical index id="M6">\begin{document}$ -\frac{3}{4} single KdV equation, indexes id="M7">\begin{document}$ -\frac{13}{12} $\end{document}</tex-math></inline-formula>, id="M8">\begin{document}$ id="M9">\begin{document}$ 0 id="M10">\begin{document}$ \frac{3}{4} $\end{document}</tex-math></inline-formula>.</p><p As result, classified into classes, each corresponds unique id="M11">\begin{document}$ s^{*}\in\{-\frac{13}{12}, \, -\frac{3}{4}, \frac{3}{4}\} such any is locally analytically if id="M12">\begin{document}$ s>s^{*} while estimate fails id="M13">\begin{document}$ s<s^{*} $\end{document}</tex-math></inline-formula>.</p>