Nonlinear PDEs with Modulated Dispersion IV: Normal Form Approach and Unconditional Uniqueness

Abstract

We study the modulated Korteweg–de Vries equation (KdV) on the circle with a time non-homogeneous modulation acting on the linear dispersion term. By adapting the normal form approach to the modulated setting, we prove sharp unconditional uniqueness of solutions to the modulated KdV in

L^2(\mathbb{T})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2(\mathbb {T})$$\end{document}

if a modulation is sufficiently irregular. For example, this result implies that if the modulation is given by a sample path of a fractional Brownian motion with Hurst index

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0< H < \frac{2}{5}$$\end{document}

, the modulated KdV on the circle is unconditionally well posed in

L^2(\mathbb{T})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2(\mathbb {T})$$\end{document}

. At a philosophical level, our normal form approach can be viewed as a controlled path approach to nonlinear Young integration, which allows for the construction of solutions to the modulated KdV (and the associated nonlinear Young integral) without assuming any positive (Hölder) regularity in time. As an interesting byproduct of our normal form approach, we obtain an improved Euler approximation scheme as compared to the classical sewing lemma approach. We also establish analogous sharp unconditional uniqueness results for the modulated Benjamin–Ono equation and the modulated derivative nonlinear Schrödinger equation (NLS) with a quadratic nonlinearity. In the appendix, we prove sharp unconditional uniqueness of the cubic modulated NLS on the circle in

H^{\frac{1}{6}}(\mathbb{T})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\frac{1}{6}}(\mathbb {T})$$\end{document}

.

Keywords

Modulated dispersion / Dispersion management / Korteweg–de Vries equation / Benjamin–Ono equation / Schrödinger equation / Normal form reduction / Unconditional uniqueness / Young integral / Regularization by noise / 35Q53 / 35Q35 / 35Q55 / 60L20 / 65M12 / 65M15

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Funding

European Research Council(no. 864138)

EPSRC Mathematical Sciences Small Grant(no. EP/Y033507/1)

UKRI Frontier Research grant(EP/Z534328/1)

Natural Science Foundation of China(no. 12501181)

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