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Group-invariant solutions of differential equations. (English) Zbl 0621.35007
The authors describe a general approach to group-invariant solutions of partial differential equations. They introduce the concept of a ”weak symmetry group” of a system of partial differential equations and show how, in principle, to construct group-invariant solutions for any group of transformations by reducing the number of variables in the system. But the paper also contains the result that every solution of a given system can be found using the reduction method with some weak symmetry group. The theoretical considerations are illustrated by a number of examples, including the heat equation, a nonlinear wave equation and a version of the Boussinesq equation.
Reviewer: W.Watzlawek

35A30Geometric theory for PDE, characteristics, transformations
35G20General theory of nonlinear higher-order PDE
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