Exact Homoclinic and Heteroclinic Solutions of the Gray--Scott Model for Autocatalysis

In this paper we obtain explicit nontrivial stationary patterns in the one-dimensional Gray--Scott model for cubic autocatalysis. Involved in the reaction are two chemicals, A and B, whose diffusion coefficients are denoted by DA and DB, respectively. The chemical A is fed into the system at a rate kf, reacts with the catalyst B at a rate k1, and the catalyst decays at a rate k2 . If these parameters obey the relation (*) kf /DA = k2 /DB, then, for appropriate values of the rate constants, we present explicit expressions for two families of pulses and one kink. We also show that if (*) is only satisfied approximately, these families of pulses are preserved, and there exists a smooth branch of kinks leading from the explicit one obtained when (*) is satisfied. We determine the local behavior of this branch near the explicit kink.