Global Well-Posedness for H^{-1}(mathbb {R}) Perturbations of KdV with Exotic Spatial Asymptotics

Abstract

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 the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022. https://doi.org/10.1088/1361-6544/ac37f5) 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. https://doi.org/10.4007/annals.2019.190.1.4) where Vequiv 0. In that setting, it is known that H^{-1}(mathbb {R}) is sharp in the class of H^s(mathbb {R}) spaces.