Centre Manifold Reduction for Quasilinear Discrete Systems

James, Michael

doi:10.1007/s00332-002-0525-x

Cited by 89 publications

(172 citation statements)

References 49 publications

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“…Here we shall consider situations when the operator L is unbounded (theory developed in [28]). We shall restrict the discussion to maps in Hilbert spaces, but the same theory could be carried out in Banach spaces (with some additional technicalities involved in the cut-off of nonlinear terms [36]).…”

Section: Finite-dimensional Casementioning

confidence: 99%

“…a space of periodic functions), which can be locally analyzed using a center manifold reduction [28]. As an example, consider the FPU lattice (6.1).…”

Section: Example In Infinite Dimensionsmentioning

confidence: 99%

“…in multicomponent Klein-Gordon lattices [28] or diatomic FPU chains [32]. In these examples, the center space is finite-dimensional and π c takes a simple form in term of Fourier coefficients.…”

Section: That the Only Part Of The Spectrum Of L In A Lies On The Unimentioning

confidence: 99%

“…In addition, the method has been extended by one of us (G.J.) to infinite-dimensional quasilinear maps in Hilbert spaces [28]. The center manifold theorem proved in this context can be used to study time-periodic oscillations in infinite oscillator chains.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical computations from references [24,34] are also presented to illustrate the mathematical results. On the other hand we give an overview of the general center manifold theorems which have been applied in this context, considering both infinitedimensional differential equations [35,36] and quasilinear maps [28]. We only examine the aspects of the theory which are relevant to the above class of nonlinear lattices.…”

Section: Introductionmentioning

confidence: 99%

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Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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Abstract. Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi-Pasta-Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete KleinGordon equations. IntroductionThe seminal work of Fermi, Pasta and Ulam in 1955 [12] had a broad impact on the development of the theory of nonlinear lattices. The Fermi-PastaUlam (FPU) model consists of a chain of masses nonlinearly coupled to their nearest neighbors. For a general nearest-neighbors interaction potential V , the governing equations readwhere we note x n the displacement of the nth mass with respect to an equilibrium position (in this version of the model the chain is of infinite extent). The anharmonic interaction potential V satisfies V (0) = 0, V (0) = 1. System (6.1) is Hamiltonian, with total energy

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Section: Finite-dimensional Casementioning

confidence: 99%

“…a space of periodic functions), which can be locally analyzed using a center manifold reduction [28]. As an example, consider the FPU lattice (6.1).…”

Section: Example In Infinite Dimensionsmentioning

confidence: 99%

Section: That the Only Part Of The Spectrum Of L In A Lies On The Unimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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Homoclinic solutions in periodic difference equations with mixed nonlinearities

Lin

Zhou

2015

Math Methods in App Sciences

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In this paper, by using critical point theory in combination with periodic approximations, we obtain some new sufficient conditions on the nonexistence and existence of homoclinic solutions for a class of periodic difference equations. Unlike the existing literatures that always assume that the nonlinear terms are only either superlinear or asymptotically linear at ∞, but superlinear at 0, our nonlinear term can mix superlinear nonlinearities with asymptotically linear ones at both ∞ and 0. To the best of our knowledge, this is the first time to consider the homoclinic solutions of this class of difference equations with mixed nonlinearities. Our results are necessary in some sense, and extend and improve some existing ones even for some special cases. Copyright © 2015 John Wiley & Sons, Ltd.

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

Giannoulis

Herrmann

Mielke

Analysis, Modeling and Simulation of Multiscale Problems

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Centre Manifold Reduction for Quasilinear Discrete Systems

Cited by 89 publications

References 49 publications

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

Homoclinic solutions in periodic difference equations with mixed nonlinearities

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

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