Continuum Descriptions for the Dynamics in Discrete Lattices: Derivation and Justification

Johannes Giannoulis (), Michael Herrmann () and Alexander Mielke ()
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Johannes Giannoulis: Weierstraür Angewandte Analysis und Stochastik
Michael Herrmann: Humboldt-Universität zu Berlin, Institut für Mathematik
Alexander Mielke: Weierstraür Angewandte Analysis und Stochastik

A chapter in Analysis, Modeling and Simulation of Multiscale Problems, 2006, pp 435-466 from Springer

Abstract: Summary The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical behavior. Depending on the initial conditions one sees quite different behavior like macroscopic elastic deformations associated with acoustic waves or like propagation of optical pulses. We show how on a formal level different macroscopic systems can be derived such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, Whitham’s modulation equation, the three-wave interaction model, or the energy transport equation using the Wigner measure. We also address the question how the microscopic Hamiltonian and the Lagrangian structures transfer to similar structures on the macroscopic level. Finally we discuss rigorous analytical convergence results of the microscopic system to the macroscopic one by either weak-convergence methods or by quantitative error bounds.

Keywords: Solitary Wave; Hamiltonian System; Invariant Manifold; Modulation Equation; Continuum Description (search for similar items in EconPapers)
Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-35657-8_16

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DOI: 10.1007/3-540-35657-6_16

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