Physikalische Mathematik

Lie symmetries and exact solutions of a class of thin lm equations

Abstract

Roman Cherniha and Liliia Myroniuk

A symmetry group classi cation for fourth-order reaction-di usion equations, allow-ing for both second-order and fourth-order di usion terms, is carried out. The fourth-order equations are treated, rstly, as systems of second-order equations that bear some resemblance to systems of coupled reaction-di usion equations with cross di usion, sec-ondly, as systems of a second-order equation and two rst-order equations. The paper generalizes the results of Lie symmetry analysis derived earlier for particular cases of these equations. Various exact solutions are constructed using Lie symmetry reductions of the reaction-di usion systems to ordinary di erential 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.