NONCLASSICAL SYMMETRY REDUCTIONS OF THE BOUSSINESQ EQUATION

In this paper we discuss symmetry reductions and exact solutions of th e Boussinesq equation using the classical Lie method of infinitesimals , the direct method due to Clarkson and Kruskal and the nonclassical m ethod due to Bluman and Cole. In particular, we compare and contrast t he application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and diffe rential Grobner bases as a technique for solving the overdetermined sy stems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exac t solutions of partial differential equations are also mentioned. To c onclude we describe some of the important open problems in the field o f symmetry analysis of differential equations.