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

Abstract

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.