Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) + H^{ -1}(\mathbb{ R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $V(0,x) \in H^5(\mathbb{ R}/\mathbb{ Z})$ satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In our companion paper we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced by Killip and Vi\c{ s}an; in the special case $V\equiv 0$, we recover their sharp $H^{ -1}(\mathbb{ R})$ result.
Categories: math.AP