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

Solutions to Nonlinear Partial Differential Equations from Symmetry-Enhancing and Symmetry-Preserving Constraints

AbstractWe show how solutions to practical partial differential equations can be found by classical symmetry reductions of a larger system of equations including constraints that result in the enlarged system having a larger symmetry group or an identical symmetry group compared to the original target equation. Symmetry-enhancing constraints provide additional similarity solutions beyond those of the standard Lie algorithm for scalar equations. Symmetry-preserving constraints enable solutions to be found more easily, after reduction of variables. Examples are given for the cylindrical boundary layer equations and for a Navier–Stokes formulation of Schrödinger wave mechanics

[SADE] A Maple package for the Symmetry Analysis of Differential Equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund and potential symmetries, invariant solutions, first-integrals, N\"other theorem for both discrete and continuous systems, solution of ordinary differential equations, reduction of order or dimension using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented in the package QPSI by the authors) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given