KdV on an incoming tide

Abstract

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 $H^{-1}$.

arXiv commentThe manuscript has been modified to better reflect the revisions made to the companion paper arXiv:2104.11346

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Cited by in corpus (2)

• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• Soliton asymptotics for KdV shock waves via classical inverse scattering