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On the Energy Distribution in Fermi–Pasta–Ulam Lattices

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Abstract

This paper presents a rigorous study, for Fermi–Pasta–Ulam (FPU) chains with large particle numbers, of the formation of a packet of modes with geometrically decaying harmonic energies from an initially excited single low-frequency mode and the metastability of this packet over longer time scales. The analysis uses modulated Fourier expansions in time of solutions to the FPU system, and exploits the existence of almost-invariant energies in the modulation system. The results and techniques apply to the FPU α- and β-models as well as to higher-order nonlinearities. They are valid in the regime of scaling between particle number and total energy in which the FPU system can be viewed as a perturbation to a linear system, considered over time scales that go far beyond standard perturbation theory. Weak non-resonance estimates for the almost-resonant frequencies determine the time scales that can be covered by this analysis.

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Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211, Geneva 4, Switzerland

    Ernst Hairer

  2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, 72076, Tübingen, Germany

    Christian Lubich

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  1. Ernst Hairer
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  2. Christian Lubich
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Correspondence to Ernst Hairer.

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Communicated by A. Mielke

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Hairer, E., Lubich, C. On the Energy Distribution in Fermi–Pasta–Ulam Lattices. Arch Rational Mech Anal 205, 993–1029 (2012). https://doi.org/10.1007/s00205-012-0526-3

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  • DOI: https://doi.org/10.1007/s00205-012-0526-3

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