In this paper we study symmetry reductions of a class of nonlinear third order partial differential equationswhere ǫ, κ, α and β are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters ǫ = 1, α = −1, β = 3 and κ = 1 2 , admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters ǫ = 0, α = 1, β = 3 and κ = 0, admits a "compacton" solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters ǫ = 1, α = −3 and β = 2, has a "peakon" solitary wave solution.A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.

In this paper we discuss the derivation of symmetry reductions and exact solutions of nonlinear partial differential equations using the classical Lie method of infinitesimal transformations, the direct method due to Clarkson and Kruskal [22], and the nonclassical method due to Bluman and Cole [11]. In particular, we compare and contrast the application of these three methods and discuss the relationships among the methods.

In this article we study various systems that represent the shallow water wave equation vxxt+αvvt−βvx∂x‐1(vt) −vt−vx = 0, where (∂x−1f)(x)=∫x∞f(y) dy, and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equationswhere α, β, γ, κ and µ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a "Boussinesqtype" equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both "compacton" and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are not obtainable through the classical method.

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