Symmetries of a class of nonlinear third-order partial differential equations

In this paper, we study symmetry reductions of a class of nonlinear third-order partial differential equations ut − ϵuxxt + 2κux = uuxxx + αuux + βuxuxx, (1) where ϵ, 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 ϵ = 1, α = −1, β = 3, and κ = 12, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters ϵ = 0, α = 1, β = 3, and κ = 0, admits a “compacton” solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters ϵ = 1, α = − 3, and β = 2, has a “peakon” solitary wave solution. logue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.