ar X iv : s ol v - in t / 9 60 90 04 v 1 1 3 Se p 19 96 Symmetries of a class of Nonlinear Third Order Partial Differential Equations

Abstract

In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations ut − ǫuxxt + 2κux = uuxxx + αuux + βuxuxx , (1) where 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.

Cite this paper

@inproceedings{Clarkson2008arXI, title={ar X iv : s ol v - in t / 9 60 90 04 v 1 1 3 Se p 19 96 Symmetries of a class of Nonlinear Third Order Partial Differential Equations}, author={Peter A. Clarkson and Elizabeth L. Mansfield and Tony Priestley}, year={2008} }