Local Well-Posedness for the Derivative Nonlinear Schr\"odinger Equation in Besov spaces

TL;DR

This paper proves local well-posedness for the cubic derivative nonlinear Schrödinger equation in Besov spaces, extending results to both non-periodic and periodic settings for initial data with regularity s ≥ 1/2.

Contribution

It establishes local well-posedness in Besov spaces for the derivative NLS, generalizing previous results from Sobolev spaces and covering both periodic and non-periodic cases.

Findings

Well-posedness in Besov spaces for s ≥ 1/2

Unified treatment of periodic and non-periodic cases

Extension of Herr's strategy to Besov spaces

Abstract

It is shown that the cubic derivative nonlinear Schr\"odinger equation is locally well-posed in Besov spaces $B^{s}_{2,\infty}(\mathbb X)$, $s\ge\tfrac12$, where we treat the non-periodic setting $\mathbb X=\mathbb R$ and the periodic setting $\mathbb X=\mathbb T$ simultaneously. The proof is based on the strategy of Herr for initial data in $H^{s}(\mathbb T)$, $s\ge\tfrac12$.