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Symmetry groups and conservation laws of certain partial differential equations. (English) Zbl 0957.35012
Summary: We study partial differential equations by using invariance under transformations groups due to Sophus Lie. This method, the so-called classical Lie method of infinitesimal transformations or symmetry groups theory, has been applied in recent years to important PDEs arising from mathematics and physics. Based on the finding of a symmetry group associated to the studied PDEs system, we can find a lot of properties related to solutions.
The aim of this paper is to point out the ideas of the Ph.D. Thesis [N. Bilă, Symmetry groups and conservation laws of certain partial differential equations, Ph.D. Thesis, University of Politehnica of Bucharest, 1999], which contains recent results obtained by application of the Lie’s method for certain PDEs systems which arise from differential geometry, especially from Ţiţeica (Tzitzeica) surfaces theory, and from physics. The paper is split into four parts: the symmetry group history, symmetry groups in differential geometry, the thesis ideas and original results thereof.
MSC:
35A30 Geometric theory, characteristics, transformations in context of PDEs
58J70 Invariance and symmetry properties for PDEs on manifolds
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