Slowly-modulated two pulse solutions and pulse splitting bifurcations

Abstract

Two pulse solutions play a central role in the phenomena of selfreplicating pulses in D reactiondiusion systems In this work we focus on the D GrayScott model as a prototype We carry out an existence and stability study for solutions consisting of two pulses moving apart from each other with slowly varying velocities In the various parameter regimes critical maximum wave speeds are identied and ODEs are derived for the wave speed and for the separation distance between the pulses The bifurcations in which these solutions are created and annihilated and in which they gain stability are determined Good agreement is found between these theoretical predictions and the results from numerical simulations The initial separation distance between the pulses need only be such that the fast activator components are exponentially small In particular there is no requirement in the analysis on the inhibitor component which is in fact far away from its homogeneous value and slowly varying between the pulses as is critical for the dynamics Hence the results presented here apply to the strong pulse interaction problem Finally we show how these results may be used to answer central questions about pulse splitting including how they provide a possible mechanism The main methods used are analytical and geometric singular perturbation theory for the existence demonstration and the nonlocal eigenvalue problem NLEP method developed in our earlier work for the stability analysis