Unconditional uniqueness for the derivative nonlinear Schr\"{o}dinger equation by normal form approach

TL;DR

This paper establishes the uniqueness of solutions to the derivative nonlinear Schrödinger equation in a critical Sobolev space using a refined normal form reduction approach that handles logarithmic divergences.

Contribution

It introduces a novel normal form reduction method tailored for the critical regularity case, overcoming previous limitations and directly proving uniqueness in $L^ olinebreak^ olinebreak ext{infty}_tH^{1/2}_x$.

Findings

Proves uniqueness of solutions in $L^ olinebreak^ olinebreak ext{infty}_tH^{1/2}_x$.

Develops a refined NFR scheme to handle logarithmic divergences.

Achieves direct uniqueness proof without relying on continuity in time.

Abstract

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.