Global well-posedness and orbital stability of solitary waves for Zakharov-Ito equation

In this paper, we consider the Zakharov-Ito equation \begin{equation*} \begin{cases} u_t+u_{xxx}+3uu_x+\rho\rho_x=0,\\ \rho_t+{(u\rho)}_x=0. \end{cases} \end{equation*} We prove the local well-posedness in $H^s\times H^s$ for $s>3/2$ and global well-posedness in $H^s\times H^s$ for $s\geq2$. When $\rho=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\in(0,+\infty)$. We prove the orbital stability of these solitary waves in $H^1\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \cite{GSS1987}.