Slowly modulated two-pulse solutions in the Gray-Scott model I: Asymptoticconstruction and stability

Two pulse solutions playa 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 consis ting of two symmetric pulses moving apart from each other with slowly varyi ng velocities. This corresponds to a "mildly strong" pulse interaction prob lem in which the inhibitor concentration varies on long spatial length scal es. Critical maximum wave speeds are identified, and ODEs are derived for t he wave speed and for the separation distance between the pulses. In additi on, the formal linear stability of these two-pulse solutions is determined. Good agreement is found between these theoretical predictions and the resu lts from numerical simulations. The main methods used in this paper are ana lytical singular perturbation theory for the existence demonstration and th e nonlocal eigenvalue problem (NLEP) method developed in our earlier work f or the stability analysis. The analysis of this paper is continued in [A. D oelman, W. Eckhaus, and T. Kaper, SIAM J. Appl. Math., to appear], where we employ geometric methods to determine the bifurcations of the slowly modul ated two-pulse solutions. In addition, in Part II we identify and quantify the central role of the slowly varying inhibitor concentration for two-puls e solutions in determining pulse splitting, and we answer some central ques tions about pulse splitting.