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Zeitschrift für Analysis und ihre Anwendungen


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Volume 36, Issue 4, 2017, pp. 437–475
DOI: 10.4171/ZAA/1596

Published online: 2017-10-09

Uniform Asymptotic Expansions for the Fundamental Solution of Infinite Harmonic Chains

Alexander Mielke[1] and Carsten Patz[2]

(1) Weierstrass Institute für Angewandte Analysis und Stochastik and Humboldt-Universität, Berlin, Germany
(2) Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany

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 particlar we give precise asymptotics for the cross-over from the $t^{-1/2}$ decay of nondegenerate wave numbers to the degenerate $t^{-1/3}$ decay of degenerate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

Keywords: Asymptotic analysis, method of stationary phase, dispersive decay, oscillatory integrals, Airy function, Fermi–Pasta–Ulam chain

Mielke Alexander, Patz Carsten: Uniform Asymptotic Expansions for the Fundamental Solution of Infinite Harmonic Chains. Z. Anal. Anwend. 36 (2017), 437-475. doi: 10.4171/ZAA/1596