Dynamics of nonlocal and local discrete Ginzburg–Landau equations: Global attractors and their congruence. (February 2022)

• Record Type: Journal Article
• Authors: Hennig, Dirk; Karachalios, Nikos I.
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: Discrete Ginzburg–Landau equation -- Discrete non-local nonlinearity -- Restricted attractor -- Global attractor -- Attractors congruence -- Dissipative solitons; 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.2021.112647
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Discrete Ginzburg–Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definition. We also prove the closeness of the solutions of the two models in the sense of a "continuous dependence on their initial data" in the l 2 metric under general conditions on the intrinsic linear gain or loss incorporated in the model. As a consequence of the closeness results, in the dissipative regime we establish the congruence of the attractors possessed by the semiflows of the non-local and of the local model respectively, for initial conditions in a suitable domain of attraction defined by the non-local system.