Symmetry in Nonlinear Mathematical Physics - 2009
Liliia Myroniuk (Lesya Ukrainka Volyn National University, Lutsk, Ukraine)
Lie symmetries and exact solutions of the generalized thin film equation
A symmetry group classification for fourth-order
reaction-diffusion equations, allowing for both second-order and
fourth-order diffusion terms, is carried out. The fourth order
equations are treated, firstly, as systems of second-order
equations that bears some resemblance to systems of coupled
reaction-diffusion equations with cross diffusion, secondly, as
systems of a of second-order equation and two first-order equations.
The paper generalizes the results of Lie symmetry analysis derived
earlier for particular cases of these equations. A wide range of
exact solutions are constructed using Lie symmetry reductions of the
to the ordinary differential equations.
The solutions include some unusual structures as well as the familiar types that regularly occur
in symmetry reductions, namely self-similar solutions, decelerating and decaying traveling waves,
and steady states.