Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale

TL;DR

This paper introduces a gauge transform for the KdV equation that improves well-posedness results in Fourier-Lebesgue spaces below the classical $H^{-1}$ threshold, using normal form reduction and algebraic cancellations.

Contribution

It develops a new gauge transform approach that extends well-posedness of KdV to lower regularity spaces beyond the $H^{-1}$-scale, independent of integrability.

Findings

Achieves sharp local well-posedness in Fourier-Lebesgue spaces with high integrability.

Extends the method to non-integrable quadratic nonlinear models.

Provides a new analytical framework for low-regularity analysis of dispersive equations.

Abstract

We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the $H^{-1}$-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities.