Wave and scattering operators for the nonlinear matrix Schrödinger equation on the half-line with a potential. (February 2023)

• Record Type: Journal Article
• Authors: Ballesteros, Miguel; Iniesta, Diego; Naumkin, Ivan; Peña, Clemente
• Is Part Of: Nonlinear analysis. Volume 227(2023)
• Publication Date: 2023-02
• Subjects: Nonlinear matrix Schrödinger equation -- General boundary conditions -- Scattering-supercritical nonlinearities -- Wave operators -- Inverse wave operators -- Scattering operator -- Generic potentials -- Exceptional potentials; 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.2022.113183
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the nonlinear matrix Schrödinger equation on the half-line with general selfadjoint boundary condition. We prove the existence of local solutions in the corresponding energy space. Moreover, we consider the scattering problem for this problem in the scattering-supercritical regime for both generic and exceptional potentials. We construct the wave, inverse wave and scattering operators in a certain neighborhood of the origin in the scattering space Σ = H 1 ∩ L 2, 1 for generic potentials in L 1, 2 + η, with η > 1 / 2, and for exceptional potentials in L 1, 3 under further assumptions on the input data. In particular, thanks to the general boundary conditions, we are able to obtain a scattering result for the nonlinear matrix Schrödinger equation on the line with a potential and point interactions.