Potential Symmetries

  • George W. Bluman
  • Sukeyuki Kumei
Part of the Applied Mathematical Sciences book series (AMS, volume 81)


As defined previously, a symmetry group of a differential equation is a group which maps any solution of the differential equation to another solution of the differential equation. In previous chapters we considered symmetries defined by infinitesimal transformations whose infinitesimals depend on independent variables, dependent variables, and derivatives of dependent variables. Such symmetries are local symmetries since at any point x the infinitesimals are determined if u(x) is sufficiently smooth in some neighborhood of x. In Chapters 5 and 6, by enlarging the classes of local symmetries admitted by given differential equations from point symmetries to contact symmetries, and, still more generally, to Lie-Bäklund symmetries, we could find more conservation laws, construct mappings to related differential equations, and determine more invariant solutions.


Wave Speed Point Symmetry Local Symmetry Infinitesimal Generator Auxiliary System 
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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • George W. Bluman
    • 1
  • Sukeyuki Kumei
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Faculty of Textile Science and TechnologyShinshu UniversityUeda, Nagano 386Japan

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