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 \be u_{tt} = \left(\kappa u + \gamma u^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. 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 ``Boussinesq-type'' 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 no} obtainable through the classical method

A precise definition of reduction of partial differential equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in (1+3) dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalization of the well-known symmetry reductions of the nonlinear wave equations.Comment: LaTeX, 21 page