Variational and stability properties of coupled NLS equations on the star graph. (November 2022)

• Record Type: Journal Article
• Authors: Cely, Liliana; Goloshchapova, Nataliia
• Is Part Of: Nonlinear analysis. Volume 224(2022)
• Publication Date: 2022-11
• Subjects: primary 35Q55 -- secondary 35Q40; Δ coupling -- Ground state -- Nonlinear Schrödinger equation -- Orbital stability; 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.2022.113056
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider variational and stability properties of a system of two coupled nonlinear Schrödinger equations on the star graph Γ with the δ coupling at the vertex of Γ . The first part is devoted to the proof of an existence of the ground state as the minimizer of the constrained energy in the cubic case. This result extends the one obtained recently for the coupled NLS equations on the line. In the second part, we study stability properties of several families of standing waves in the case of a general power nonlinearity. In particular, we consider one-component standing waves e i ω t ( Φ 1 ( x ), 0 ) and e i ω t ( 0, Φ 2 ( x ) ) . Moreover, we study two-component standing waves e i ω t ( Φ ( x ), Φ ( x ) ) for the case of power nonlinearity depending on a unique power parameter p . To our knowledge, these are the first results on variational and stability properties of coupled NLS equations on graphs.