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arXiv · 2104.00338

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

Dirk Hennig·Nikos I. Karachalios
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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.

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Dirk Hennig, Nikos I. Karachalios. 2021-04-01. Dynamics of nonlocal and local discrete Ginzburg-Landau equations: global attractors and their congruence. https://doi.org/10.1016/j.na.2021.112647

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