Dynamics of nonlocal and local discrete Ginzburg-Landau equations: global attractors and their congruence

TL;DR

This paper investigates the long-term behavior of discrete Ginzburg-Landau equations with local and non-local nonlinearities, establishing conditions for attractor existence, their congruence, and solution dependence on initial data.

Contribution

It introduces a detailed analysis of asymptotic behaviors, attractor properties, and solution closeness between local and non-local discrete Ginzburg-Landau models.

Findings

Identification of distinct asymptotic regimes for non-local DGL.

Proof of attractor congruence between local and non-local models in dissipative regimes.

Establishment of continuous dependence of solutions on initial data in $l^2$ metric.

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.