KdV on an incoming tide

Given smooth step-like initial data $V(0,x)$ on the real line, we show that the Korteweg–de Vries equation is globally well-posed for initial data $u(0,x) \in V(0,x) + H^{-1}($\mathbb{R}$)$. The proof uses our general well-posedness result for exotic spatial asymptotics. As a prerequisite, we show that KdV is globally well-posed for $H^3(\mathbb{R})$ perturbations of step-like initial data. In the case $V\equiv 0$, we obtain a new proof of the Bona–Smith theorem using the low-regularity methods that established the sharp well-posedness of KdV in $H^{-1}$.