Dynamics of nonlocal and local discrete Ginzburg–Landau equations: Global attractors and their congruence
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-di...
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Veröffentlicht in: | Nonlinear analysis 2022-02, Vol.215, p.112647, Article 112647 |
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Sprache: | eng |
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Zusammenfassung: | 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 l2 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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ISSN: | 0362-546X 1873-5215 |
DOI: | 10.1016/j.na.2021.112647 |