Equivalence transformations for classes of differential equations

We consider classes C of differential equations characterized by the presence of arbitrary elements, that is, arbitrary functions or constants. Based on an idea of Ovsiannikov, we develop a systematic theory of equivalence transformations, that is, point changes of variables which map every equation in C to another equation in C. Examples of nontrivial groups of equivalence transformations are found for some linear wave and nonlinear diffusion convection systems, and used to clarify some previously known results. We show how equivalence transformations may be inherited as symmetries of equations in C, leading to a partial symmetry classification for the class C. New symmetry results for a potential system form of the nonlinear diffusion convection equation are derived by this procedure. Finally we show how to use equivalence group information to facilitate complete symmetry classification for a class of differential equations. The method relies on the geometric concept of a moving frame, that is, an arbitrary (possibly noncommuting) basis for differential operators on the space of independent and dependent variables. We show how to choose a frame which is invariant under the action of the equivalence group, and how to rewrite the determining equations for symmetries in terms of this frame. A symmetry classification algorithm due to Reid is modified to deal with the case of noncommuting operators. The result is an algorithm which combines features of Reid’s classification algorithm and Cartan’s equivalence method. The method is applied to the potential diffusion convection example, and yields a complete symmetry classification in a particularly elegant form.

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