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In this paper we discuss the derivation of symmetry reductions and exact solutions of nonlinear partial differential equations using the classical Lie method of infinitesimal transformations, the direct method due to Clarkson and Kruskal [22], and the nonclassical method due to Bluman and Cole [11]. In particular, we compare and contrast the application of these three methods and discuss the relationships among the methods.

The classical, direct, and nonclassical methods for symmetry reductions of nonlinear partial differential equations