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

Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander

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

Cited by 14 publications

(17 citation statements)

References 53 publications

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“…Concerning nonlinear wave interactions, there exist several results (mostly three-wave mixing) in the context of strictly hyperbolic systems, see, e.g., [18,19,22,25], and in the case of microscopically discrete dynamical systems, cf. [11,12] and the references given therein. For spatially periodic systems we again refer to [27] and to [1], where a similar system of coupled-mode equations is derived from Maxwell's equations in photonic crystals.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Giannoulis

Mielke

Sparber

2008

Journal of Differential Equations

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Giannoulis

Mielke

Sparber

2008

Journal of Differential Equations

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“…Concerning Σ red we observe that the ansatz (3.39) can be written as As mentioned in the introduction, one can obtain the macroscopic equations (1.17) also by inserting the two-scale ansatz (1.16) into the Klein-Gordon chain (3.2) and requiring the coefficients of the terms ε 2 e i(ωnt+θnj) to vanish. Based on this formal expansion one can then justify the validity of (1.17), see [Gia06,Gia08] and §7.2 in [GHM06]. These equalities reflect the cancelation in L 0 and manifest the equipartition of energy for plane-wave solutions.…”

Section: Three-wave-interaction For the Kg Chainmentioning

confidence: 84%

“…However, for all examples presented here we discuss the corresponding justification problem after having derived the reduced Lagrangian and Hamiltonian structures. We also refer to the surveys [Mie02,GHM06,SU07] and to [Mie08] for an abstract theory using Γ-convergence for Hamiltonian systems.…”

Section: General Approach To Lagrangian and Hamiltonian Two-scale Redmentioning

confidence: 99%

“…In other words, within Whitham's modulation theory we do not know the two-scale transformations a priori and this complicates the reduction of Lagrangian and Hamiltonian structures. Finally, the modulation theory for FPU chains leads to further complications, since the Galilean invariance of (3.1) causes a coupling between macroscopic waves and modulated oscillations, see [FV99,Her05,GHM06,DHR06,DH07].…”

Section: Outlook To Further Examplesmentioning

confidence: 99%

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Lagrangian and Hamiltonian two-scale reduction

Giannoulis

Herrmann

Mielke

2008

Journal of Mathematical Physics

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Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system.In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions.In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

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“…The proof is based on methods developed for dispersive PDE's (cf. [7]) and the main ideas are outlined in the review article [6]. A key ingredient are sharp decay rates for the linearized system.…”

Section: Introductionmentioning

confidence: 99%

On dispersive stability of Hamiltonian systems on lattices

Patz¹

2007

Proc Appl Math and Mech

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We study the long-time dynamics of oscillations in lattices of infinitely many particles interacting via certain non-linear potentials. The aim is to proof dispersive stability of such Hamiltonian systems analogously to results known for PDEs. To do so we first recapitulate the dynamics of linear Hamiltonian systems on an infinite chain and give optimal decay rates based on the dispersion relation. Based on this we proof that if the non-linearity is weak enough, the non-linear system shows a similar behaviour like its linearization. *

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

Cited by 14 publications

References 53 publications

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Lagrangian and Hamiltonian two-scale reduction

On dispersive stability of Hamiltonian systems on lattices

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