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The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems

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Abstract

The interaction of stable pulse solutions on R 1 is considered when distances between pulses are sufficiently large. We construct an attractive local invariant manifold giving the dynamics of interacting pulses in a mathematically rigorous way. The equations describing the flow on the manifold is also given in an explicit form. By it, we can easily analyze the movement of pulses such as repulsiveness, attractivity and/or the existence of bound states of pulses. Interaction of front solutions are also treated in a similar way.

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REFERENCES

  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.

    Google Scholar 

  2. Carr, J., and Pego, R. L. (1989). Metastable patterns in solutions of ut=ε2uxx+f (u).Comm. Pure Appl. Math.42, 523-576.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  6. Evans, J. W. (1972). Nerve axon equations: II Stability at rest. Indiana Univ. Math. J. 22, 75-90.

    Google Scholar 

  7. Evans, J. W. (1972). Nerve axon equations: III Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577-593.

    Google Scholar 

  8. Evans, J. W. (1975). Nerve axon equations: IV The stable and unstable impulses. Indiana Univ. Math. J. 24, 1169-1190.

    Google Scholar 

  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. 2n-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.

    Google Scholar 

  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.

    Google Scholar 

  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 FitzHugh_Nagumo system. Trans. AMS 286, 431-469.

    Google Scholar 

  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. 5, 335-361.

    Google Scholar 

  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.

    Google Scholar 

  21. Kawasaki, K. and Ohta, T. (1982). Kink dynamics in one-dimensional nonlinear systems. Physica A 116, 573-593.

    Google Scholar 

  22. Landau, L. D., and Lifshitz, E. M. (1977). Quantum Mechanics, Pergamon, Oxford.

    Google Scholar 

  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.

    Google Scholar 

  24. Nishiura, Y., and Ueyama, D. (1999). A skeleton structure of self-replicating dynamics. Physica D 130, 73-104.

    Google Scholar 

  25. Pearson, J. E. (1993). Complex patterns in a simple system. Science 216, 189-192.

    Google Scholar 

  26. Sandstede, B. (1998). Stability of multiple-pulse solutions. Trans. Amer. Math. Soc. 350, 429-472.

    Google Scholar 

  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.

    Google Scholar 

  31. Yamada, H., and Nozaki, K. (1990). Interaction of pulses in dissipative systems. Prog. Theo. Physics 84, 801-809.

    Google Scholar 

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  1. Graduate School of Integrated Science, Yokohama City University, Yokohama, 236-0027, Japan

    Shin-Ichiro Ei

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  1. Shin-Ichiro Ei
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Ei, SI. The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems. Journal of Dynamics and Differential Equations 14, 85–137 (2002). https://doi.org/10.1023/A:1012980128575

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