Advertisement

Journal of Dynamics and Differential Equations

, Volume 26, Issue 3, pp 405–459 | Cite as

Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

  • Adam Boden
  • Karsten Matthies
Article

Abstract

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in \(\mathbb {R}^2\) with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.

Keywords

Travelling waves Periodic media Spatial dynamics  Reaction-diffusion Conley index Homogenisation 

Notes

Acknowledgments

Financial support by EPSRC through DTA funding for Boden is gratefully acknowledged.

References

  1. 1.
    Afendikov, A., Mielke, A.: Bifurcations of Poiseuille flow between parallel plates: three-dimensional solutions with large spanwise wavelength. Arch. Ration. Mech. Anal. 129, 101–127 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures, AMS Chelsea Publishing, Providence, RI, Corrected reprint of the 1978 original (2011)Google Scholar
  3. 3.
    Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Math. (N.S.) 22, 1–37 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boden, A.: Travelling waves in heterogeneous media. Ph.D. thesis, University of Bath (2012)Google Scholar
  6. 6.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1999)zbMATHGoogle Scholar
  7. 7.
    Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI (1978)zbMATHGoogle Scholar
  8. 8.
    Eckmann, J.-P., Wayne, C.E.: Propagating fronts and the center manifold theorem. Commun. Math. Phys. 136, 285–307 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fiedler, B., Scheel, A.: Spatio-temporal dynamics of reaction–diffusion patterns. Trends in Nonlinear Analysis. Springer, Berlin (2003)Google Scholar
  10. 10.
    Fisher, R.A.: The advance of advantageous genes. Ann. Eugenics 7, 335–369 (1937)Google Scholar
  11. 11.
    Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical systems, Universitext. Springer-Verlag London Ltd., London (2011)CrossRefGoogle Scholar
  12. 12.
    Haragus, M., Schneider, G.: Bifurcating fronts for the Taylor–Couette problem in infinite cylinders. Z. Angew. Math. Phys. 50, 120–151 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). Translated from the Russian by G.A. YosifianGoogle Scholar
  14. 14.
    Kirchgässner, K.: Wave-solutions of reversible systems and applications. J. Differ. Equ. 45, 113–127 (1982)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kolmogorov, A., Petrovsky, I., Piskunov, N.: Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. Bull. Univ. Etat. Moscow Ser., Int. Math. Mec. Sect. A 1, 1–29 (1937)Google Scholar
  16. 16.
    Matthies, K., Schneider, G., Uecker, H.: Exponential averaging for traveling wave solutions in rapidly varying periodic media. Math. Nachr. 280, 408–422 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Matthies, K., Wayne, C.E.: Wave pinning in strips. Proc. R. Soc. Edinb. Sect. A 136, 971–995 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol. 59, 2nd edn. Springer, New York (2007)zbMATHGoogle Scholar
  20. 20.
    Sandstede, B., Scheel, A.: Defects in oscillatory media: toward a classification. SIAM J. Appl. Dyn. Syst 3, 1–68 (2004)Google Scholar
  21. 21.
    Schneider, G., Uecker, H.: Existence and stability of modulating pulse solutions in Maxwell’s equations describing nonlinear optics. Z. Angew. Math. Phys. 54, 677–712 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Smoller, J.: Shock Waves and Reaction–Diffusion Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  23. 23.
    Taylor, M.E.: Partial Differential Equations. III. Applied Mathematical Sciences, vol. 117. Springer, New York (1997)Google Scholar
  24. 24.
    Vanderbauwhede, A., Iooss, G.: Center Manifold Theory in Infinite Dimensions, vol. 1, pp. 125–163. Springer, Berlin (1992)Google Scholar
  25. 25.
    Xin, J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)Google Scholar
  26. 26.
    Xin, J.: An Introduction to Fronts in Random Media. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 5. Springer, New York (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations