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

Roman Cherniha,Liliia Myroniuk

doi:10.4303/jpm/p100508

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 bear some resemblance to systems of coupled reaction-diffusion equations with cross diffusion, secondly, as systems of a 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. Various exact solutions are constructed using Lie symmetry reductions of the reaction-diffusion systems to 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.

Highlights

We consider the fourth-order nonlinear partial differential equation (PDE) of the form: ut = − K(u)uxxx x + D(u)ux x + F (u), (1.1)where K, D, and F are arbitrary smooth functions
We reduce systems arising in case 3 of Table 1 to systems of ordinary differential equations (ODEs); these ODE systems are solved in particular cases, and exact solutions of the initial PDE systems are constructed
Equation (1.1) has been treated, firstly, as a system of secondorder equations that bears some resemblance to a system of coupled reaction-diffusion equations with cross diffusion, secondly, as a system of a second-order equation and two first-order equations

Summary

Introduction

Equations (2.14)–(2.16) with R(t) = P (t) = 0 and nonzero K(u) are equivalent to ξt0(t) = 4α = 0, D(u) = 0, F (u) = 0, and K(u) is an arbitrary smooth function This means that the triplet (K(u), 0, 0) forms the system from case 1 of Table 1 and the coordinates of the infinitesimal operator (2.4) take the form: ξ0 = 4αt + t0, ξ1 = αx + x0, η1 = η2 = 0,. Substituting the functions R(t), P (t), and ξ0 into (2.13), we obtain the infinitesimal operator (2.4), which generates three basic operators listed in case 3 of Table 1. One notes that cases 2 and 3 of Table 1 generalize the results of Lie symmetry analysis for the Cahn-Hilliard equation derived in [22, 30]. We note that the determining equations obtained (without equation (2.63)) are equivalent to those (2.7)–(2.12) for the system (2.1), so that no new Lie point symmetries or contact symmetries can be found

Symmetry reduction and exact solutions

Findings

Conclusions