Unconditional uniqueness for the derivative nonlinear Schr\"odinger equation on the real line

TL;DR

This paper establishes the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation on the real line at near-critical regularity levels, using a normal form transformation and energy estimates.

Contribution

It introduces a normal form approach to prove uniqueness for DNLS solutions at low regularity without auxiliary spaces.

Findings

Proves unconditional uniqueness in $H^s(\ eal)$ for $s > 1/2$.

Transforms DNLS into a normal form equation for easier analysis.

Shows low-regularity solutions satisfy the normal form equation via $H^{s-1}$ estimates.

Abstract

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schr\"odinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in $H^s(\mathbb{R})$, $s>\frac12$, without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the $H^{s-1}(\mathbb{R})$-norm.