Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model I: Asymptotic Construction and Stability

Abstract

Two pulse solutions play a central role in the phenomena of self-replicating pulses in the one-dimensional (1-D) Gray--Scott model. In the present work (part I of two parts), we carry out an existence study for solutions consisting of two symmetric pulses moving apart from each other with slowly varying velocities. This corresponds to a "mildly strong" pulse interaction problem in which the inhibitor concentration varies on long spatial length scales. Critical maximum wave speeds are identified, and ODEs are derived for the wave speed and for the separation distance between the pulses. In addition, the formal linear stability of these two-pulse solutions is determined. Good agreement is found between these theoretical predictions and the results from numerical simulations. The main methods used in this paper are analytical singular perturbation theory for the existence demonstration and the nonlocal eigenvalue problem (NLEP) method developed in our earlier work for the stability analysis. The analysis of this paper is continued in [A. Doelman, W. Eckhaus, and T. Kaper, SIAM J. Appl. Math., to appear], where we employ geometric methods to determine the bifurcations of the slowly modulated two-pulse solutions. In addition, in Part II we identify and quantify the central role of the slowly varying inhibitor concentration for two-pulse solutions in determining pulse splitting, and we answer some central questions about pulse splitting.