Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 066, 11 pages      arXiv:1105.5303
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction

Stephen C. Anco a, Sajid Ali b and Thomas Wolf a
a) Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1 Canada
b) School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology, H-12 Campus, Islamabad 44000, Pakistan

Received March 05, 2011, in final form July 03, 2011; Published online July 12, 2011; Typos in the solutions are corrected August 02, 2013

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.

Key words: semilinear heat equation; similarity reduction; exact solutions; group foliation; symmetry.

pdf (342 kb)   tex (14 kb)       [previous version:  pdf (323 kb)   tex (13 kb)]


  1. Ovsiannikov L.V., Group analysis of differential equations, Academic Press, Inc., New York - London, 1982.
  2. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
  3. Bluman G., Anco S.C., Symmetry and integration methods for differential equations, Applied Mathematical Sciences, Vol. 154, Springer-Verlag, New York, 2002.
  4. Anco S.C., Liu S., Exact solutions of semilinear radial wave equations in n dimensions, J. Math. Anal. Appl. 297 (2004), 317-342, math-ph/0309049.
  5. Anco S.C., Ali S., Wolf T., Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions, J. Math. Anal. Appl. 379 (2011), 748-763, arXiv:1011.4633.
  6. Dorodnitsyn V.A., On invariant solutions of the equation of nonlinear heat conduction with a source, USSR Comp. Math. Math. Phys. 22 (1982), 115-122.
  7. Galaktionov V.A., Svirshchevskii S.R., Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, Chapman & Hall/CRC, Boca Raton, 2007.
  8. Wolf T., Applications of CRACK in the classification of integrable systems, in Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, Vol. 37, Amer. Math. Soc., Providence, RI, 2004, 283-300,
  9. Clarkson P.A., Mansfield E.L., Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys. D 70 (1993), 250-288, solv-int/9306002.
  10. Arrigo D.J., Hill J.M., Broadbridge P., Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source, IMA J. Appl. Math. 52 (1994), 1-24.
  11. Vijayakumar K., On the integrability and exact solutions of the nonlinear diffusion equation with a nonlinear source, J. Austral. Math. Soc. Ser. B 39 (1998), 513-517.
  12. Qu C., Zhang S.-L., Group foliation method and functional separation of variables to nonlinear diffusion equations, Chinese Phys. Lett. 22 (2005), 1563-1566.
  13. Golovin S.V., Applications of the differential invariants of infinite dimensional groups in hydrodynamics, Commun. Nonlinear Sci. Numer. Simul. 9 (2004), 35-51.
  14. Nutku Y., Sheftel M.B., Differential invariants and group foliation for the complex Monge-Ampère equation, J. Phys. A: Math. Gen. 34 (2001), 137-156.
  15. Martina L., Sheftel M.B., Winternitz P., Group foliation and non-invariant solutions of the heavenly equation, J. Phys. A: Math. Gen. 34 (2001), 9243-9263, math-ph/0108004.
  16. Sheftel M.B., Method of group foliation and non-invariant solutions of partial differential equations. Example: the heavenly equation, Eur. Phys. J. B Condens. Matter Phys. 29 (2002), 203-206.

Previous article   Next article   Contents of Volume 7 (2011)