On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$

TL;DR

This paper proves local well-posedness for the periodic Korteweg-de Vries equation in Sobolev spaces with regularity above -2/3, extending previous results and avoiding reliance on complete integrability.

Contribution

It introduces a modulation restricted normal form method to establish well-posedness below the previous threshold of -1/2, providing a new analytical approach for KdV.

Findings

Established local well-posedness for s > -2/3 in H^s(a0T).

Extended the known regularity threshold below -1/2.

Developed a novel normal form approach for KdV.

Abstract

We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s> -\frac23$. This work creates an analogue of the mKdV result by Nakanishi, Takaoka, and Tsutsumi for KdV, extending the currently best-known result of $s \geq -\frac12$ without utilizing the theory of complete integrability.