Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics

TL;DR

This paper establishes global well-posedness for KdV with initial data close to a solution V(t,x) in H^{-1}, allowing for exotic spatial asymptotics and including localized and periodic perturbations.

Contribution

It extends well-posedness results for KdV to initial data with irregular asymptotics and includes a broader class of perturbations around solutions V(t,x).

Findings

Proves global well-posedness for initial data in V(0,x)+H^{-1}( R).

Includes localized and periodic perturbations of KdV solutions.

Employs commuting flows method to achieve results.

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 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.