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 $u_t-\epsilon u_{xxt}+2\kappa u_x=u{u}_{xxx}+\alpha u{u}_x+\beta u_x{u}_{xx}$ where $\epsilon$, $\kappa$, $\alpha$ and $\beta$ 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 $\epsilon =1$, $\alpha =-1$, $\beta =3$ 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 $\epsilon =0$, $\alpha =1$, $\beta =3$ and $\kappa =0$, admits a ``compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters $\epsilon =1$, $\alpha =-3$ and $\beta =2$, 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.

arXiv comment22 pages, tex, Mathematical and Computer Modelling (to appear)