The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems

Journal of Dynamics and Differential Equations

Volumn 14, Issue 1, 2002, Pages 85-137

  Ei, Shin Ichiro ^a  

^a YOKOHAMA CITY UNIVERSITY   (Japan)

Author keywords

Interaction of fronts; Interaction of pulses; Reaction diffusion systems

Indexed keywords

EID: 0006030342     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1023/A:1012980128575     Document Type: Article

References (31)

1. Alikakos, N., Bates, P. W., and Fusco, G. (1991). Slow motion for the Cahn-Hilliard equation in one space dimension. J. Diff. Eq. 90, 81-135.
2. xx +f(u). Comm. Pure Appl. Math. 42, 523-576.
3. Doelman, A., Gardner, R. A., and Kaper, T. J. (1998). Stability analysis of singular patterns in the 1-D Gray-Scott model. Physica D 122, 1-36.
4. Ei, S.-I. (1994). Interaction of pulses in FitzHugh-Nagumo equations. In Reaction-Diffusion Equations and Their Applications and Computational Aspects, China-Japan Symposium (Li, T.-T., Mimura, M, Nishiura, Y., Ye, Q-X., eds.), World Scientific, pp. 6-13.
5. Evans, J. W. (1972). Nerve axon equations: I Linear approximations. Indiana Univ. Math. J. 21, 877-955.
6. Evans, J. W. (1972). Nerve axon equations: II Stability at rest. Indiana Univ. Math. J. 22, 75-90.
7. Evans, J. W. (1972). Nerve axon equations: III Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577-593.
8. Evans, J. W. (1975). Nerve axon equations: IV The stable and unstable impulses. Indiana Univ. Math. J. 24, 1169-1190.
9. Ei, S.-I., Ikeda, H., and Yanagida, E., On the eigenfunctions of the adjoint operator associated with a pulse solution of the FitzHugh-Nagumo systems, in preparation.
10. Ei, S.-I., Mimura, M., and Nagayama, M. Pulse-pulse interaction in reaction-diffusion systems, preprint.
11. Ei, S.-I., Nishiura, Y., and Sandstede, B. Pulse-interaction approach to self-replicating dynamics in reaction-diffusion systems, in preparation.
12. Ei, S.-I., Nishiura, Y., and Ueda, K. 2"-Splitting or edge-splitting?, to appear in Japan J. Ind. Appl. Math.
13. Ei, S.-I., and Ohta, T. (1994). The motion of interacting pulses, Phys. Rev. E 50(6), 4672-4678.
14. Ei, S.-I., and Wei, J. Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt system in 2-dimensional space, preprint.
15. Fusco, G., and Hale, J. (1989). Slow motion manifold, dormant instability and singular perturbations. J. Dynam. Diff. Eqs. 1, 75-94.
16. Gorshkov, K. A., and Ostrovsky, L. A. (1981). Physica 3D.
17. Jones, C. K. R. T. (1984). Stability of the traveling wave solution of the FitzHughNagumo system. Trans. AMS 286, 431-4169.
18. Fife, P. C., and Mcleod, J. B. (1977). The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335-361.
19. Henry, D. (1981). Geometric theory of semilinear parabolic equations, Lecture Notes in Math., Vol. 840, Springer-Verlag.
20. Kan-on, Y. (1995). Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Appl. Math. 26, 340-363.
21. Kawasaki, K. and Ohta, T. (1982). Kink dynamics in one-dimensional nonlinear systems. Physica A 116, 573-593.
22. Landau, L. D., and Lifshitz, E. M. (1977). Quantum Mechanics, Pergamon, Oxford.
23. Nii, S. (1999). A topological proof of stability of multiple-front solutions of the FitzHugh-Nagumo equations. J. Dynam. Diff. Eqs. 11, 515-555.
24. Nishiura, Y., and Ueyama, D. (1999). A skeleton structure of self-replicating dynamics. Physica D 130, 73-104.
25. Pearson, J. E. (1993). Complex patterns in a simple system. Science 216, 189-192.
26. Sandstede, B. (1998). Stability of multiple-pulse solutions. Trans. Amer. Math. Soc. 350, 429-4172.
27. Sandstede, B. Weak interaction of pulses, in preparation.
28. Schatzmann, M., in preparation.
29. Tazaki, I., and Matsumoto, G. (1975). Mechanism of nerve impulses, Sangyo Tosyo (in Japanese).
30. Yanagida, E. (1985). Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations. J. Math. Biol. 22, 81-104.
31. Yamada, H., and Nozaki, K. (1990). Interaction of pulses in dissipative systems. Prog. Theo. Physics 84, 801-809.