Stable bound states of pulses in an excitable medium

The interaction of one-dimensional pulses is studied in the excitable regim e of a two variable reaction-diffusion model. The model is capable of exhib iting long range attraction of pulses and formation of stable bound pulse s tates. The important features of pulse interactions can be captured by a co mbination of various analytical and numerical methods. A kinematic ansatz t reating pulses as particle-like interacting structures is described. Their interaction is determined using the dispersion relation for pulse trains, w hich gives the dependence of the speed c(d) of the wavetrain on its wavelen gth d. Anomalous dispersion for large d, i.e. a negative slope of c(d), cor responds to long range pulse attraction. Stable bound pairs are possible if the medium exhibits long range attraction and there is at least one maximu m of the dispersion curve. We compare predictions of the kinematic theory w ith numerical simulations and stability analysis. If the slope of the dispe rsion curve changes sign, branches of non-equidistant pulse train solutions bifurcate and may lead to bound pulse states. The transition from normal l ong range dispersion, typical in excitable media, to the anomalous dispersi on studied here can be understood through a multiscale perturbation theory for pulse interactions. We derive the relevant equations, which yield an an alytic expression for non-monotonic dispersion curves with a finite number of extrema. (C) 2000 Elsevier Science B.V. All rights reserved.