Advertisement

Journal of Statistical Physics

, Volume 161, Issue 4, pp 801–820 | Cite as

An Exactly Solvable Travelling Wave Equation in the Fisher–KPP Class

  • Éric BrunetEmail author
  • Bernard Derrida
Article

Abstract

For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times \(t_n\) at which the travelling wave reaches the positions n, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher–KPP class can be recovered, in particular Bramson’s shifts of the positions. We also recover and generalize Ebert–van Saarloos’ corrections depending on the initial condition.

Keywords

Fisher–KPP Front equation Travelling wave 

References

  1. 1.
    Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve propagation. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1975)Google Scholar
  2. 2.
    McKean, H.P.: Applications of brownian motion to the equation of Kolmogorov–Petrovski–Piscounov. Commun. Pure Appl. Math. 28, 323 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Kametaka, Y.: On the nonlinear diffusion equation of Kolmogorov–Petrovskii–Piskunov type. Osaka J. Math. 13, 11 (1976)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bramson, M.D.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bramson, M.D.: Convergence of solutions of the Kolmogorov equation to traveling waves. Mem. Am. Math. Soc. 44, 1–190 (1983)MathSciNetGoogle Scholar
  6. 6.
    Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses and traveling waves. J. Stat. Phys. 51, 817 (1988)zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Murray, J.D.: Mathematical biology I: an introduction. Interdisciplinary Applied Mathematics, 3rd edn. Springer, New York (2002)Google Scholar
  8. 8.
    Meerson, B., Vilenkin, A., Sasorov, P.V.: Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts. Phys. Rev. E 87(1), 012117 (2013)CrossRefADSGoogle Scholar
  9. 9.
    Munier S.: Lecture notes on quantum chromodynamics and statistical physics. arXiv:1410.6478 (2014)
  10. 10.
    Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355 (1937)CrossRefGoogle Scholar
  11. 11.
    Kolmogorov A, Petrovsky I., Piscounov N.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, A 1, 1 (1937)Google Scholar
  12. 12.
    Ma, W., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-linear Mech. 31, 329 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Ablowitz, M.J., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29 (2003)zbMATHCrossRefADSGoogle Scholar
  15. 15.
    Benguria, R.D., Depassier, M.C.: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Commun. Math. Phys. 175, 221 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Hamel, F., Nolen, J., Roquejoffre, J.M., Ryzhik, L.: A short proof of the logarithmic Bramson correction in Fisher–KPP equations. NHM 8, 275 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Ebert, U., van Saarloos, W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Phys. D 146, 1 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Mueller, A.H., Munier, S.: Phenomenological picture of fluctuations in branching random walks. Phys. Rev. E 90(4), 042143 (2014)MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Brunet, É., Derrida, B., Mueller, A.H., Munier, S.: A phenomenological theory giving the full statistics of the position of fluctuating pulled fronts. Phys. Rev. E 73, 056126 (2006)CrossRefADSGoogle Scholar
  20. 20.
    Mueller, C., Mytnik, L., Quastel, J.: Effect of noise on front propagation in reaction-diffusion equations of KPP type. Invent. math. 184, 405 (2010)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Henderson C.: Population stabilization in branching Brownian motion with absorption. arXiv:1409.4836 (2014)

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.LPS-ENS, UMR 8550, CNRSUPMC Univ Paris 06, Sorbonne UniversitésParisFrance
  2. 2.Collège de FranceParisFrance

Personalised recommendations