Bulletin of Mathematical Biology

, Volume 41, Issue 6, pp 835–840 | Cite as

Explicit solutions of Fisher's equation for a special wave speed

  • Mark J. Ablowitz
  • Anthony Zeppetella


The travelling waves for Fisher's equation are shown to be of a simple nature for the special wave speeds\(c = \pm 5/\sqrt {(6)} \). In this case the equation is shown to be of Painlevé type, i.e. solutions admit only poles as movable singularities. The general solution for this wave speed is found and a method is presented that can be applied to the solution of other nonlinear equations of biological and physical interest.


Wave Speed Laurent Series Physical Interest Simple Nature Exponential Series 
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  1. Ablowitz, M. J., A. Ramani, and H. Segur. 1978. Lettre at Nuovo Cimento, accepted for publication.Google Scholar
  2. Abramowitz, M. and I. Stegun. 1965.Handbook of Mathematical Functions. New York: Dover.Google Scholar
  3. Canosa, J. 1969. “Diffusion in Nonlinear Multiplicative Media.”J. Math. Phys.,10, 1862.CrossRefGoogle Scholar
  4. Fife, P. C. 1978. “Asymptotic States for Equations of Reaction and Diffusion.”.Bull Am. Math. Soc.,84, 693–726.zbMATHMathSciNetCrossRefGoogle Scholar
  5. Fisher, R. A. 1936. “The Wave of Advance of an Advantageous Gene.”Ann. Eugen. 7, 355–369.Google Scholar
  6. Hoppensteadt, F. C. 1975.Mathematical Theories of Populations: Demographics, Genetics and Epidemics. CMBS Vol. 20, Philadelphia: SIAM Publication.zbMATHGoogle Scholar
  7. Ince, E. I. 1956.Ordinary Differential Equations. New York: Dover.Google Scholar
  8. Kolmogorov, A., I. Petrovskii and N. Piskunov 1937. “Étude de l'Équation de la Diffusion avec Croissance de la Quantité de la Matière et Son Application à un Problème Biologique.”Moscow Univ. Bull. Math.,1, 1–25.zbMATHGoogle Scholar
  9. Lefschetz, S. 1963:Differential Equations: Geometric Theory, 2nd ed., pp. 95–111. New York: Interscience.zbMATHGoogle Scholar
  10. McKean, H. P.. 1975. “Application of Brownian Motion to the Equation of Kolmogorov-Petrovskii-Piskunov.”Communs Pure appl. Math. 28, 323–331.zbMATHMathSciNetGoogle Scholar
  11. Zeppetella, A. and M. J. Ablowitz. 1978. “On Solutions of Some Nonlinear Ordinary Differential Equations with Applications to Fisher's Equations.” To be submitted toSIAM J. Appl. Math. Google Scholar

Copyright information

© Society for Mathematical Biology 1979

Authors and Affiliations

  • Mark J. Ablowitz
    • 1
  • Anthony Zeppetella
    • 1
  1. 1.Department of Mathematics and Computer ScienceClarkson CollegePotsdamUSA

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