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

Abstract

AbstractGiven a suitable solution V(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data $$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ u ( 0 , x ) ∈ V ( 0 , x ) + H - 1 ( R ) . Our conditions on V do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ V ( 0 , x ) ∈ H 5 ( 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. 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 $$V\equiv 0$$ V ≡ 0 . In that setting, it is known that $$H^{-1}(\mathbb {R})$$ H - 1 ( R ) is sharp in the class of $$H^s(\mathbb {R})$$ H s ( R ) spaces.