IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-94-011-2050-0_2.html

Internal Symmetries of Differential Equations

In: Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics

Author

Listed:
  • Ian Anderson

    (Utah State University, Department of Mathematics)

  • Niky Kamran

    (McGill University, Department of Mathematics)

  • Peter J. Olver

    (University of Maryland, Department of Mathematics
    University of Minnesota, School of Mathematics)

Abstract

Bäcklund’s Theorem, which characterizes contact transformations, is generalized to give an analogous characterization of “internal symmetries” of systems of differential equations. For a wide class of systems of differential equations, every internal symmetry comes from a first order generalized symmetry and, conversely, every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry. We analyze the contact conditions in detail, deducing powerful necessary conditions for a system of differential equations admit “genuine” internal symmetries, i.e., ones which do not come from classical “external” symmetries. Applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to Hilbert and Cartan are the noncompact real form of the exceptional simple Lie group G 2.

Suggested Citation

  • Ian Anderson & Niky Kamran & Peter J. Olver, 1993. "Internal Symmetries of Differential Equations," Springer Books, in: N. H. Ibragimov & M. Torrisi & A. Valenti (ed.), Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, pages 7-21, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-2050-0_2
    DOI: 10.1007/978-94-011-2050-0_2
    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:978-94-011-2050-0_2. 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.