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 react ion are two chemicals, A and B, whose diffusion coefficients are denoted by D-A and D-B, respectively. The chemical A is fed into the system at a rate k(f), reacts with the catalyst B at a rate k(1), and the catalyst decays a t a rate k(2). If these parameters obey the relation (*) k(f)/D-A = k(2)/D-B, then, for appropriate values of the rate constants, we present explicit exp ressions for two families of pulses and one kink. We also show that if (*) is only satisfied approximately, these families of pulses are preserved, an d there exists a smooth branch of kinks leading from the explicit one obtai ned when (*) is satisfied. We determine the local behavior of this branch n ear the explicit kink.