Similarity reductions of partial differential equations

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Abstract
The author determine the relation between two methods of reduction, the similarity reduction method using nonclassical symmetry groups, and the direct approach used by Clarkson and Kruskal (1989). They prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of Clarkson and Kruskal. The two procedures are equivalent only if the generators zeta (x,t,u), tau (x,t,u), eta (x,t,u) of the nonclassical groups are such that zeta / tau is independent of u. To explain these results, he proves the existence of families of solutions of the Burgers equation which are found by means of nonclassical symmetry reduction, and which cannot be determined via the general reduction form of Clarkson and Kruskal.