Non-classical symmetries and the singular manifold method: the Burgers and the Burgers-Huxley equations

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Abstract
In this paper a generalization of the direct method of Clarkson and Kruskal (1989) for finding similarity reductions of partial differential equations is found and discussed for the Burgers and Burgers-Huxley equations. The generalization incorporates the singular manifold method largely based upon the Painleve property. This singular manifold can be used as a reduced variable. Furthermore, a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the PDE. This procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable. The symmetries found in this way are shown to be those corresponding to the so-called non-classical symmetries by Bluman and Cole (1974), and Olver and Rosenau (1986).