Journal of Physical Mathematics

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

Roman Cherniha and Liliia Myroniuk

Full-text: Open access


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.

Article information

J. Phys. Math., Volume 2 (2010), Article ID P100508, 19 pages.

First available in Project Euclid: 25 October 2010

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 35K50 35K60: Nonlinear initial value problems for linear parabolic equations

Lie symmetries exact solutions thin film equations


Cherniha, Roman; Myroniuk, Liliia. Lie symmetries and exact solutions of a class of thin film equations. J. Phys. Math. 2 (2010), Article ID P100508, 19 pages. doi:10.4303/jpm/P100508.

Export citation


  • R. Anderson and N. Ibragimov. Lie-Bäcklund Transformations in Applications. SIAM Studies in Applied Mathematics 1, SIAM, Philadelphia, 1979.
  • D. Arrigo and J. Hill. Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math., 94 (1995), 21–39.
  • D. Arrigo, J. Hill, and P. Broadbridge. Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source. IMA J. Appl. Math., 52 (1994), 1–24.
  • F. Bernis, J. Hulshof, and J. King. Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity, 13 (2000), 413–439.
  • F. Bernis and J. McLeod. Similarity solutions of a higher order nonlinear diffusion equation. Nonlinear Anal., 17 (1991), 1039–1068.
  • A. Bertozzi. The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc., 45 (1998), 689–697.
  • A. Bertozzi and M. Pugh. Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math., 51 (1998), 625–661.
  • G. Bluman and J. Cole. Similarity Methods for Differential Equations. Springer, Berlin, 1974.
  • B. Bradshaw-Hajek and P. Broadbridge. A robust cubic reaction-diffusion system for gene propagation. Math. Comput. Modelling, 39 (2004), 1151–1163.
  • P. Broadbridge. Entropy diagnostics for fourth order partial differential equations in conservation form. Entropy, 10 (2008), 365–379.
  • K. Brown and A. Lacey. Reaction-Diffusion Equations. Oxford University Press, New York, 1990.
  • M. Bruzón, M. Gandarias, E. Medina, and E. Muriel. New symmetry reductions for a lubrication model. In “Nonlinear Physics: Theory and Experiment, II”. M. J. Ablowitz, M. Boiti, F. Pempinelli, and B. Prinari (Eds.), World Scientific, Singapore, (2002), pp. 143–148.
  • J. Cahn and J. Hilliard. Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28 (1958), 258–266.
  • J. Cahn and J. Taylor. Surface motion by surface diffusion. Acta Metall. Mater., 42 (1994), 1045–1063.
  • R. Cantrell and C. Cosner. Spatial Ecology via Reaction-Diffusion Equations. John Wiley & Sons, Chichester, 2003.
  • R. Cherniha and J. King. Lie symmetries and conservation laws of non-linear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math., 71 (2006), 391–408.
  • R. Cherniha and J. King. Lie symmetries of nonlinear multidimensional reaction-diffusion systems. I. J. Phys. A, 33 (2000), 267–282, 7839–7841.
  • R. Cherniha and J. King. Lie symmetries of nonlinear multidimensional reaction-diffusion systems. II. J. Phys. A, 36 (2003), 405–425.
  • R. Cherniha and M. Serov. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms. European J. Appl. Math., 9 (1998), 527–542.
  • R. Cherniha and M. Serov. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms. II. European J. Appl. Math., 17 (2006), 597–605.
  • R. Cherniha, M. Serov, and I. Rassokha. Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations. J. Math. Anal. Appl., 342 (2008), 1363–1379.
  • S. Choudhury. General similarity reductions of a family of Cahn-Hilliard equations. Nonlinear Anal., 24 (1995), 131–146.
  • P. Clarkson and E. Mansfield. Symmetry reductions and exact solutions of a class of nonlinear heat equations. Phys. D, 70 (1994), 250–288.
  • V. Dorodnitsyn. On invariant solutions of non-linear heat conduction with a source. USSR Comput. Math. and Math. Phys., 22 (1982), 115–122.
  • T. Dupont, R. Goldstein, L. Kadanoff, and S.-M. Zhou. Finite-time singularity formation in Hele-Shaw systems. Phys. Rev. E (3), 47 (1993), 4182–4196.
  • J. Evans, V. Galaktionov, and J. King. Blow-up similarity solutions of the fourth-order unstable thin film equation. European J. Appl. Math., 18 (2007), 195–231.
  • J. Evans, V. Galaktionov, and J. Williams. Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation. SIAM J. Math. Anal., 38 (2006), 64–102.
  • W. Fushchich, W. Shtelen, and M. Serov. Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer, Dordrecht, 1993.
  • V. Galaktionov and S. Svirshchevskii. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.
  • M. Gandarias and M. Bruzón. Symmetry analysis and solutions for a family of Cahn-Hilliard equations. Rep. Math. Phys., 46 (2000), 89–97.
  • M. Gandarias and N. Ibragimov. Equivalence group of a fourth-order evolution equation unifying various non-linear models. Comm. in Nonlin. Sci. and Num. Simulation, 13 (2008), 259–268.
  • P. Gaskell, P. Jimack, M. Sellier, and H. Thompson. Flow of evaporating, gravity-driven thin liquid films over topography. Phys. Fluids, 18 (2006), 013601.1–013601.14.
  • H. P. Greenspan. On the motion of a small viscous droplet that wets a surface. J. Fluid Mech., 84 (1978), 125–143.
  • M. Grinfeld and A. Novick-Cohen. The viscous Cahn-Hilliard equation: morse decomposition and structure of the global attractor. Trans. Amer. Math. Soc., 351 (1999), 2375–2406.
  • J. R. King. Mathematical analysis of a model for substitutional diffusion. Proc. Roy. Soc. London Ser. A, 430 (1990), 377–404.
  • J. G. Kingston and C. Sophocleous. On form-preserving point transformations of partial differential equations, J. Phys. A, 31 (1998), 1597–1619.
  • W. Mullins. Theory of thermal grooving. J. Appl. Phys., 28 (1957), 333–339.
  • J. D. Murray. Mathematical Biology. Springer, Berlin, 1989.
  • A. Novick-Cohen and L. Segel. Nonlinear aspects of the Cahn-Hilliard equation. Phys. D, 10 (1984), 277–298.
  • P. Olver. Applications of Lie Groups to Differential Equations. Springer, Berlin, 1986.
  • A. Oron, S. Davis, and S. Bankoff. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69 (1997), 931–980.
  • L. Ovsiannikov. Group Analysis of Differential Equations. Academic Press, New York, 1982
  • L. Ovsyannikov. Group relations of the equation of non-linear heat conductivity, Dokl. Akad. Nauk SSSR, 125 (1959), 492–495.
  • A. Polyanin and V. Zaitsev. Handbook of Exact Solutions for Ordinary Differential Equations. Chapman & Hall/CRC, Boca Raton, FL, 2003.
  • N. Smyth and J. Hill. High-order nonlinear diffusion. IMA J. Appl. Math., 40 (1988), 73–86.
  • U. Valbusa, C. Boragno, and F. Buatier de Mongeot. Nanostructuring surfaces by ion sputtering, J. Phys.: Condens. Matter, 14 (2002), 8153–8175.
  • O. O. Vaneeva, A. G. Johnpillai, R. O. Popovych, and C. Sophocleous, Enhanced group analysis and conservation laws of variable coefficient reactiondiffusion equations with power nonlinearities. J. Math. Anal. Appl., 330 (2007), 1363–1386.
  • K. Vu, J. Butcher, and J. Carminati. Similarity solutions of partial differential equations using DESOLV. Comput. Phys. Comm., 176 (2007), 682–693. }