Symmetries of a class of nonlinear fourth order partial differential equations

摘要

In this paper we study symmetry reductions of a class of nonlinear fourthorder partial differential equations \be u_{tt} = \left(\kappa u + \gammau^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,\ee where $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ are constants. Thisequation may be thought of as a fourth order analogue of a generalization ofthe Camassa-Holm equation, about which there has been considerable recentinterest. Further equation (1) is a ``Boussinesq-type'' equation which arisesas a model of vibrations of an anharmonic mass-spring chain and admits both``compacton'' and conventional solitons. A catalogue of symmetry reductions forequation (1) is obtained using the classical Lie method and the nonclassicalmethod due to Bluman and Cole. In particular we obtain several reductions usingthe nonclassical method which are no} obtainable through the classical method.