WIAS Preprint No. 1846, (2013)

Uniform asymptotic expansions for the infinite harmonic chain



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Patz, Carsten

2010 Mathematics Subject Classification

  • 37K60 41A60 42B20

Keywords

  • Asymptotic analysis, oscillatory integrals, Fermi-Pasta-Ulam chain, Airy function, dispersive decay, method of stationary phase

Abstract

We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

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