Lie symmetries and exact solutions of the generalized thin film equation

Abstract:
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 reaction-diffusion systems 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.