A compactness result for inhomogeneous nonlinear Schrödinger equations. (February 2022)

• Record Type: Journal Article
• Authors: Dinh, Van Duong; Keraani, Sahbi
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: 35Q55; Inhomogeneous nonlinear Schrödinger equation -- Compactness property -- Linear profile decomposition -- Nonlinear profile decomposition; 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.2021.112617
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We establish a compactness property of the difference between nonlinear and linear operators (or the Duhamel operator) related to the inhomogeneous nonlinear Schrödinger equation. The proof is based on a refined profile decomposition for the equation. More precisely, we prove that any sequence ( ϕ n ) n of H 1 -functions which converges weakly in H 1 to a function ϕ, the corresponding solutions with initial data ϕ n can be decomposed (up to a remainder term) as a sum of the corresponding solution with initial data ϕ and solutions to the linear equation.