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 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 Vişan; in the special case $V\equiv 0$, we recover their sharp $H^{-1}(\mathbb{R})$ result.

The hypotheses and proof of Theorem 1.2 have been updated in response to the comments of the associate editor

References in corpus (6)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations
• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• KdV equation beyond standard assumptions on initial data
• Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications
• KdV on an incoming tide

Cited by in corpus (2)

• KdV on an incoming tide
• Soliton asymptotics for KdV shock waves via classical inverse scattering