WIAS Preprint No. 1132, (2006)

Wave trains, solitons and modulation theory in FPU chains



Authors

  • Dreyer, Wolfgang
  • Herrmann, Michael
  • Rademacher, Jens D. M.

2010 Mathematics Subject Classification

  • 34E13 37K60 70F45 70K70 82C21

Keywords

  • FPU chain, traveling waves, multiscale ansatz, modulation theory, dispersive shocks

Abstract

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.

Appeared in

  • Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 467--500

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