Three-Dimensional Waves in Excitable Reaction-Diffusion Systems: the Eikonal Approximation

  • J. Gomatam
  • P. Grindrod
Part of the NATO ASI Series book series (NSSB, volume 244)


There is a wealth of experimental observation on pattern formation in three dimensional Belousov-Zhabotinsky (BZ) reagent as documented by Welsh [1, 2] and Winfree [3]. The diversity of wave forms evolving in the reagent highlights the complexity of the task faced by theoreticians working within the framework of the full reaction-diffusion (R-D) equations with excitable kinetics. This points to the need for developing analytically tractable, abbreviated models which will at least capture the geometry of the wave forms. An approach that offers considerable scope for generalization to three-dimensional R-D systems is a geometric theory (the eikonal method) proposed by Keener [4] and Zykov [5] in the context of spiral waves on a plane.


Wave Form Spiral Wave Eikonal Equation Eikonal Approximation Helical Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Welsh, B.J. (1984). Pattern Formation in the Belousov- Zhabotinsky Reaction. Ph.D. Thesis: Glasgow College.Google Scholar
  2. [2]
    Welsh, B.J., Gomatam, J. & Burgess, A.E. (1983). Three- dimensional chemical waves in the Belousov-Zhabotinsky reaction. Nature 304, 611–614.ADSCrossRefGoogle Scholar
  3. [3]
    Winfree, A.T. (1987). When time breaks down: the three- dimensional dynamics of electrochemical waves and cardiac arrhythmias. Princeton University Press.Google Scholar
  4. [4]
    Keener, J.P. (1986). A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46, 1039–1056.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Zykov, V.S. (1988). Simulation of wave processes in excitable media; translation by A.T. Winfree, Manchester Univesity Press: Manchester.Google Scholar
  6. [6]
    Grindrod, P. & Gomatam, J. (1987). The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds. J. Math. Biol. 25, 597–610.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Maselko, J. & Showalter, K. (1989). Chemical waves on spherical surfaces. Nature 339, 609–611.ADSCrossRefGoogle Scholar
  8. [8]
    Showalter, K. (1989). Private communication.Google Scholar
  9. [9]
    Gomatam, J. & Grindrod, P. (1987). Three-dimensional waves in excitable reaction-diffusion systems. J. Math. Biol. 25, 611–622.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Grindrod, P. Patterns and waves in reaction diffusion. Oxford University Press, to appear.Google Scholar
  11. [11]
    Gomatam, J. (1982). Pattern synthesis from singular solutions in the Debye limit; helical waves and twisted toroidal scroll structures. J. Phys. A; Math. Gen. 15, 1463–1476.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Winfree, A.T. & Strogatz, S.H. (1984). Singular filaments organize chemical waves in three-dimensions: 4 wave taxonomy. Physica 13D, 221–233.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • J. Gomatam
    • 1
  • P. Grindrod
    • 2
  1. 1.Department of MathematicsGlasgow CollegeGlasgowUK
  2. 2.INTERA-ECLHighlands FarmHenley-on-Thames, OxonUK

Personalised recommendations