Multiple-peak traveling waves of the Gray-Scott model

Abstract

We study a reaction-diffusion system which models the pre-mixed isothermal autocatalytic chemical reaction of order m  (m > 1) between two chemical species, a reactant A and an auto-catalyst B,   A + m B  --> (m+1) B, and a linear decay B -->C, where  C is an inert product. The special case of m = 2 is the much studied Gray-Scott model, but without feeding. We prove existence of multiple traveling waves which have distinctive number of local maximaor peaks. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses than related models in literature such as isothermal autocatalysis without decay, or a bio-reactor model with isothermal autocatalysis of order m + 1 with m-th order of decay.