Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations utt = ( κu+ γu ) xx + uuxxxx + μuxxtt + αuxuxxx + βu 2 xx , (1) where 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.