×

Pulse-pulse interaction in reaction-diffusion systems. (English) Zbl 1098.35542

Summary: It had been long believed that one-dimensional travelling pulses and the corresponding two-dimensional expanding rings and spiral waves arising in excitable reaction-diffusion systems annihilate when they closely approach one another. However, recently it has been numerically confirmed that if the velocity is very slow, expanding rings and spiral do not necessarily annihilate. In particular, in some situation, two closely approaching pulses reflect, as if they were elastic like objects. By using the center manifold theory, we show that if there are travelling pulses which primarily and super-critically bifurcate from a standing pulse when some parameter is varied, they possess reflection mechanism if the velocity is very slow.

MSC:

35K57 Reaction-diffusion equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eqs. 14 (1) (2002) 85-137.; S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eqs. 14 (1) (2002) 85-137. · Zbl 1007.35039
[2] Gray, P.; Scott, S. K., Autocatalytic reaction in the isothermal continuous stirred tank reactor, Chem. Eng. Sci., 39, 1087-1097 (1984)
[3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981.; D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981. · Zbl 0456.35001
[4] Kawaguchi, S.; Mimura, M., Collision of travelling waves in a reaction-diffusion system with global coupling effect, SIAM J. Appl. Math., 59, 920-941 (1998) · Zbl 0936.35089
[5] Krisher, K.; Mikhailov, A., Bifurcation to travelling spots in reaction-diffusion systems, Phys. Rev. Lett., 73, 3165-3168 (1994)
[6] Mimura, M.; Nagayama, M., Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability, Chaos, 7, 4, 817-826 (1997) · Zbl 0933.35094
[7] Petrov, V.; Scott, S. K.; Showalter, K., Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion systems, Philos. Trans. R. Soc. London A, 347, 631-642 (1994) · Zbl 0867.35047
[8] Schenk, C. P.; Or-Guil, M.; Bode, M.; Purwins, H.-G., Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78, 3781-3784 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.