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  1. Abstract

    Given a suitable solutionV(tx) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$u(0,x)V(0,x)+H-1(R). Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$V(0,x)H5(R/Z)satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022. we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019. where$$V\equiv 0$$V0. In that setting, it is known that$$H^{-1}(\mathbb {R})$$H-1(R)is sharp in the class of$$H^s(\mathbb {R})$$Hs(R)spaces.

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