Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations

Pulse-pulse interactions play central roles in a variety of pattern formation phenomena, including self-replication. In this article, we develop a theory for the semistrong interaction of pulses in a class of singularly perturbed coupled reaction-diffusion equations that includes the (generalized) Gierer--Meinhardt, Gray--Scott, Schnakenberg, and Thomas models, among others. Geometric conditions are determined on the reaction kinetics for whether the pulses in a two-pulse solution attract or repel, and ODEs are derived for the time-dependent separation distance between their centers and for their wave speeds. In addition, conditions for the existence of stationary two-pulse solutions are identified, and the interactions between stationary and dynamically evolving two-pulse solutions are studied. The theoretical results are illustrated in a series of examples. In two of these, which are related to the classical Gierer--Meinhardt equation, we find that the pulse amplitudes blow up in finite time. Moreover, the blowup of stationary one-pulse solutions and of dynamically varying two-pulse solutions occurs precisely at the parameter values for which the theory we develop predicts that these solutions should cease to exist as bounded solutions. Finally, generalizations to N-pulse solutions are presented.