Existence and nonexistence of travelling waves of a reaction–diffusion system

In this paper we study the existence and nonexistence of travelling waves of the reaction–diffusion system { a t = a x x − E a x − a b − k a b q , b t = D b x x − D E b x + a b + k a b q , where a , b are concentration densities of the two chemical species, D > 0 is the ratio of diffusion coefficients of the two chemical species involved, E is the strength of the electric field applied, k > 0 is a parameter and q > 1 . The system arises from the study of two-dimensional iodate-arsenous-acid (IAA) reaction with a constant electric field applied. (IAA) reaction is very much studied in experiments and demonstrates the interesting phenomenon of instability driven by buoyancy of travelling waves. By using a novel approach, explicit bounds c ∗ ( D , E ) and c ∗ ( D , E ) are derived such that there is a unique travelling wave of every speed c ≥ c ∗ ( D , E ) and there does not exist any travelling wave of speed c < c ∗ ( D , E ) , with q being fixed. In addition, c ∗ ( D , E ) bears a close resemblance of the minimum travelling wave speed of the classical case of D = 1 and E = 0 , when 0 < E ≪ 1 and D ∼ 1 .