Nonlinear PDEs with modulated dispersion IV: normal form approach and unconditional uniqueness

TL;DR

This paper develops a normal form approach to prove unconditional uniqueness for modulated KdV, Benjamin-Ono, and NLS equations with irregular modulation, extending solution construction without regularity assumptions and improving integral approximation methods.

Contribution

It introduces a novel normal form method for modulated PDEs, establishing unconditional uniqueness under irregular modulation and extending nonlinear Young integral construction.

Findings

Proves unconditional uniqueness for modulated KdV with fractional Brownian motion modulation.

Extends nonlinear Young integral to broader function classes and improves Euler approximation.

Establishes sharp unconditional uniqueness results for modulated Benjamin-Ono and NLS equations.

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 $L^2(\mathbb T)$ 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 $0 < H < \frac 25$, the modulated KdV on the circle is unconditionally well-posed in $L^2(\mathbb T)$. Our normal form approach provides the construction of solutions to the modulated KdV (and the associated nonlinear Young integral) {\it without} assuming any positive regularity in time. As an interesting byproduct of our normal form approach, we extend the construction of the nonlinear Young integral to a much larger class of functions, and 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 Schr\"odinger equation (NLS) with a quadratic nonlinearity. In the appendix, we prove sharp unconditional uniqueness of the cubic modulated NLS on the circle in $H^{\frac 16}(\mathbb T)$.