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

DOI10.1007/S00220-022-04522-7arXiv2104.11346WikidataQ122901669 ScholiaQ122901669MaRDI QIDQ6365926

Thierry Laurens

Publication date: 22 April 2021

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 $u(0,x)inV(0,x)+H^{-1}(mathbbR)$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $V(0,x)in{H}^5(mathbbR/mathbbZ)$ 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 Vic{s}an; in the special case $Vequiv0$, we recover their sharp $H^{-1}(mathbbR)$ result.

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) KdV equations (Korteweg-de Vries equations) (35Q53) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Perturbations in context of PDEs (35B20) Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems (37K40) Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems (37K15) Traveling wave solutions (35C07) Soliton solutions (35C08) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02)

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