Method of group foliation and non-invariant solutions of partial differential equations |
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Authors: | MB Sheftel |
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Institution: | (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, Russia, TR |
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Abstract: | 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.
Received 13 September 2001 Published online 2 October 2002
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ID="a"e-mail: sheftel@gursey.gov.tr |
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Keywords: | 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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