The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions

This article studies propagating wave fronts of a reaction-diffusion system modeling an isothermal chemical reaction A + 2B -> 3B involving two chemical species, a reactant A and an auto-catalyst B, whose diffusion coefficients, D-A and D-B, are unequal due to different molecular weights and/or sizes. Explicit bounds c(*) and c(*) that depend on D-B/D-A are derived such that there is a unique travelling wave of every speed c >= c* and there does not exist any travelling wave of speed c < c(*). Furthermore, the reaction-diffusion system of the Gray-Scott model of A + 2B -> 3B, and a linear decay B -> C, where C is an inert product is also studied. The existence of multiple traveling waves which have distinctive number of local maxima or peaks is shown. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses.