ABSTRACT
In this chapter we discuss a simplification of the Belousov-Zhabotinskii reaction model described in Section 4.4. In particular, we give a qualitative analysis of the front of certain travelling concentration waves which have been observed frequently in experiments. To begin with, we assume that the wave front depends primarily on the con-
centrations of bromous acid (HBrO2), which we have denoted by X , and the bromide ion (Br−) denoted by Y and to a lesser extent on the concentration of the oxidised state Ce(IV ) denoted by Z. We also assume that the diffusion coefficients DX , DY are constant and that DX = DY = D. Furthermore, for simplicity, we consider only one space dimension x. Therefore, if we neglect the concentration Z and take note of the above
assumptions, the reaction-diffusion system (4.5.9)–(4.5.11) reduces to
∂X
∂t = k1AY − k2XY + k3AX − 2k4X2 +D∂
2X
∂x2 ,
∂Y
∂t = −k1AY − k2XY +D∂
2Y
∂x2 . (8.1.1)
For later purposes it is convenient to nondimensionalise (8.1.1) by setting
u = 2k4X
k3A , v =
k2Y
k3Ar ,
x′ = ( k3A
D
)1/2 x, t′ = k3At,
L = 2k4k1 k2k3
, M = k1 k3 , b =
k2 2k4
, (8.1.2)
where r is a suitable parameter. The reason for making these transformations is that the solutions u and v, which are of interest, lie in the interval 0≤u, v≤ 1. With the transformations (8.1.2), the system (8.1.1) takes the form
∂u
∂t′ = Lrv + u(1− u− rv) + ∂
2u
∂x′2 ,
∂v
∂t′ = Mv − buv + ∂
2v
∂x′2 .
