SIMILARITY REDUCTIONS OF PARTIAL-DIFFERENTIAL EQUATIONS

We determine the relation between two methods of reduction, the similarity reduction method using non-classical symmetry groups, and the direct approach used by Clarkson and Kruskal. We prove that the solutions which are obtained by similarity in correspondence to non-classical 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 xi(x, t, u), tau(x, t, u), eta(x, t, u) of the non-classical groups are such that xi/tau is independent of u. To explain these results, we prove the existence of families of solutions of the Burgers' equation which are found by means of non-classical symmetry reduction, and which cannot be determined via the general reduction form of Clarkson and Kruskal.