Abstract |

**Optimal System and Group Invariant
Solutions for a Nonlinear Wave Equation**

*G.Maluleke, D.P.Mason (School of
Computational and Applied Mathematics, University of the Witwatersrand,
Johannesburg, South Africa )*

*e-mail:
gmalulek@cam.wits.ac.za*

We
consider a mathematical model proposed by Scott and Stevenson [1984 and 1986]
and independently by McKenzie [1984] to explain how the segregation or
migration occurs. The mathematical model is one-dimensional and it is in the
form of a third order nonlinear partial differential equation describing the
two-phase fluid flow of a medium compacting under gravity. The partial
differential equation contains two parameters, n and m, which are exponents in
power laws relating the permeability of the medium and the viscosity of the
solid matrix to the voidage. We derive
the one dimensional optimal system of the partial differential equation by
using techniques developed by Olver [1986], Coggeshall and Meyer-ter-vehn
[1992] and Chupakhin [1994]. The one dimensional optimal system is then used to
classify group invariant solutions of the differential equation.

Section
: 12