Method of group foliation and non-invariant solutions of partial differential equations

Example: the heavenly equation
  • M.B. Sheftel


Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, i.e. the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation.

PACS. 02.20.Tw Infinite-dimensional Lie groups – 02.30.Jr Partial differential equations – 04.20.Jb Exact solutions – 03.65.Fd Algebraic methods 


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Copyright information

© EDP Sciences, Springer-Verlag 2002

Authors and Affiliations

  • M.B. Sheftel
    • 1
  1. 1.Feza Gürsey Institute PO Box 6, Cengelkoy, 81220 Istanbul, Turkey and Department of Higher Mathematics, North Western State Technical University, Millionnaya Str. 5, 191186, St. Petersburg, RussiaTR

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