Local well-posedness for the nonlocal derivative nonlinear Schrödinger equation in Besov spaces. (October 2019)

• Record Type: Journal Article
• Authors: Barros, Vanessa; de Moura, Roger; Santos, Gleison
• Is Part Of: Nonlinear analysis. Volume 187(2019)
• Page Start: 320
• Page End: 338
• Publication Date: 2019-10
• Subjects: Nonlocal derivative nonlinear Schrödinger equation -- Local well-posedness -- Besov spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.05.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the Cauchy problem associated with the one-dimensional integro-differential nonlocal derivative nonlinear Schrödinger equation in the Besov space B 2 1 2, 1 ( R ) . The local well-posedness for small initial data in B 2 1 2, 1 ( R ) is established. Our method of proof combines the contraction principle applied to the associated integral equation together with interpolations of some smoothing effects (Kato's smoothing effects, Strichartz estimate and estimates for the maximal function) for phase localized functions associated to the linear dispersive part of the equation, and a fractional vector-valued Leibniz's rule derived by Molinet and Ribaud in (2004).