IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-3-540-35657-8_17.html
   My bibliography  Save this book chapter

Wave Trains, Solitons and Modulation Theory in FPU Chains

In: Analysis, Modeling and Simulation of Multiscale Problems

Author

Listed:
  • 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)

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.

Suggested Citation

  • Wolfgang Dreyer & Michael Herrmann & Jens D. M. Rademacher, 2006. "Wave Trains, Solitons and Modulation Theory in FPU Chains," Springer Books, in: Alexander Mielke (ed.), Analysis, Modeling and Simulation of Multiscale Problems, pages 467-500, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-35657-8_17
    DOI: 10.1007/3-540-35657-6_17
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-3-540-35657-8_17. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.