Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

Razvan Mosincat; Haewon Yoon

doi:10.3934/dcds.2020003

Article Contents

2020, Volume 40, Issue 1: 47-80. Doi: 10.3934/dcds.2020003

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Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

Razvan Mosincat^1, and
Haewon Yoon^2,

1.
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK
2.
National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 10617, Taiwan

Received: September 30, 2018

Published: October 2019

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Abstract

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger 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.

Keywords:

Derivative nonlinear Schrödinger equation,
unconditional well-posedness,
normal form method.

Mathematics Subject Classification: 35Q55.

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