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 $u_t -εu_{xxt} +2κu_x= u u_{xxx} +αu u_x +βu_x u_{xx}$ 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 $κ=\tfr12$, 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. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.