Nonclassical symmetry reductions of the Boussinesq equation

In this paper we discuss symmetry reductions and exact solutions of the Boussinesq equation using the classical Lie method of infinitesimals, the direct method due to Clarkson and Kruskal and the nonclassical method due to Bluman and Cole. In particular, we compare and contrast the application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and differential Gröbner bases as a technique for solving the overdetermined systems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exact solutions of partial differential equations are also mentioned. To conclude we describe some of the important open problems in the field of symmetry analysis of differential equations.