Wave Trains, Solitons and Modulation Theory in FPU Chains

Wolfgang Dreyer (), Michael Herrmann () and Jens D. M. Rademacher ()
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Wolfgang Dreyer: Weierstraß-Institut für Angewandte Analysis und Stochastik
Michael Herrmann: Humboldt-Universität zu Berlin, Institut für Mathematik
Jens D. M. Rademacher: Weierstraß-Institut für Angewandte Analysis und Stochastik

A chapter in Analysis, Modeling and Simulation of Multiscale Problems, 2006, pp 467-500 from Springer

Abstract: Summary We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.

Keywords: Wave Train; Riemann Problem; Modulation Theory; Atomic Chain; Young Measure (search for similar items in EconPapers)
Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-35657-8_17

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DOI: 10.1007/3-540-35657-6_17

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