Abstract
In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.
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S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems. Physica D.,165 (2002), 176–198.
S.-I. Ei, H. Ikeda and T. Kusaka, The dynamics of interacting fronts in reaction-diffusion systems with bistable nonlinearity, in preparation.
P. Gridiron, Pattern and waves. Oxford University Press, New York, 1991.
H. Ikeda and T. Ikeda, Bifurcation phenomena from standing pulse solutions in some reaction-diffusion systems. J. Dynam. Diff. Eqs.,12 (2000), 117–167.
H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems. SIAM J. Appl. Math.,49 (1989), 515–538.
H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems. Nonlinear Anal.,13 (1989), 507–526.
A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations. Nonlinearity,7 (1994), 805–835.
A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities. Chaos,4 (1994), 477–484.
A. Hagberg, E. Meron, I. Rubinstein and B. Maltzman, Order parameter equations for front bifurcations: planar and circular fronts. Phys. Rev. E.,55 (1997), 4450–4457.
S.P. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations. Quart. J. Math. Oxford,27 (1976), 123–134.
H. Kokubu, Y. Nishiura and H. Oka, Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems. J. Diff. Eqs.,86 (1990), 260–341.
J.D. Murray, Mathematical biology. Springer-Verlag, Berlin, 1989.
Y. Nishiura, M. Mimura, H. Ikeda and H. Fujii, Singular limit analysis of stability of travelling wave solutions in bistable reaction-diffusion systems. SIAM J. Math. Anal.,21 (1990), 85–122.
Y. Nishiura, Y. Oyama and K.-I. Ueda, Dynamics of traveling pulses in heterogeneous media of jump type. Hokkaido Math. J.,36 (2007), 207–242.
P. Ortoleva and J. Ross, Theory of propagation of discontinuities in kinetic systems with multiple time scales: fronts, front multiplicity, and pulses. J. Chem. Phys.,63 (1975), 3398–3408.
J.P. Pauwelussen, Nerve impulse propagation in a branching nerve system: a simple model. Physica D.,4 (1981), 67–88.
J.P. Pauwelussen, One way traffic pulses in a neuron. J. Math. Biology,15 (1982), 151–171.
J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system. SIAM J. Appl. Math.,42 (1982), 1111–1137.
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Ei, S.I., Ikeda, H. & Kawana, T. Dynamics of front solutions in a specific reaction-diffusion system in one dimension. Japan J. Indust. Appl. Math. 25, 117 (2008). https://doi.org/10.1007/BF03167516
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DOI: https://doi.org/10.1007/BF03167516