Well posedness for Hirota-Satsuma's equation

Abstract

We are interested in the initial-value problem associated to the Hirota-Satsuma equation in the real line $$ u_t+u_x-2uu_t+2u_x\int_x^{\infty}u_tdx'-u_{txx}=0, \quad x \in \mathbb R, $$ where $u$ is a real-valued function. This equation models the unidirectional propagation of shallow water waves as the well-known Korteweg-de Vries and Benjamin-Bona-Mahony equations. Here we show local well posedness for initial data in the space $$ \Omega_s=\{\phi \in H^s(\mathbb R) : -1 \notin \sigma(-\partial_x^2-2\phi)\} \ \text{if}\ s>\tfrac12, $$ and small initial data in $H^s(\mathbb R)$ if $0\le s \le \frac12$. We also prove global well posedness for small energy data in $H^1(\mathbb R)$.