Algorithms for the Nonclassical Method of Symmetry Reductions

In this article the authors first present an algorithm for calculating the determining equations associated with so-called “nonclassical method ” of symmetry reductions (à la Bluman and Cole) for systems of partial differential equations. This algorithm requires significantly less computation time than that standardly used, and avoids many of the difficulties commonly encountered. The proof of correctness of the algorithm is a simple application of the theory of Grobner bases. In the second part they demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear PDEs. The authors take as their principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be nonintegrable, the authors obtain an exact “two-soliton” solution from a nonclassical reduction.