# Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation

## TL;DR

This paper establishes local well-posedness for the periodic derivative nonlinear Schrödinger equation in Fourier-Lebesgue spaces for s>0, extending previous results and employing a deterministic approach inspired by stochastic PDE techniques.

## Contribution

It improves the known regularity threshold for local well-posedness of the periodic derivative NLS, closing the gap in subcritical theory with a novel deterministic method.

## Key findings

- Proves local well-posedness in Fourier-Lebesgue spaces for s>0.
- Develops a new analysis of solution structure and nonlinear submanifold construction.
- Extends previous results from s>1 to s>0, closing the subcritical gap.

## Abstract

We prove local well-posedness for the periodic derivative nonlinear Schrodinger's equation, which is L^2 critical, in Fourier-Lebesgue spaces which scale like H^s(T) for s>0. In particular we close the existing gap in the subcritical theory by improving the result of Grunrock and Herr [25], which established local well-posedness in Fourier-Lebesgue spaces which scale like H^s(T) for s>1 . We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space. Together these allow us to construct the unique solution to the given subcritical data. This constructive procedure is inspired by the theory of para-controlled distributions developed by Gubinelli-Imkeller-Perkowski [26] and Cantellier-Chouk [10] in the context of stochastic PDE. Our proof and results however, are purely deterministic.