Turing Instabilities in Reaction-Cattaneo Systems

As a second approach, we use reaction-Cattaneo equations, see Sect. 2.2.2, to study the effect of inertia on Turing instabilities [206]. The uniform steady state of the two-species reaction-Cattaneo system. 

We assume again that (Pu, Pv) satisfies the stability conditions (10.23). To determine the stability of (10.60) against spatially inhomogeneous perturbations, we set 

Linearizing (10.58a) and (10.58b) about the uniform steady state, we obtain 

The appropriate spatial modes for the fluxes 8J and 8Jy are sin jr), and we obtain the characteristic equation for the reaction-Cattaneo system, det J = 0, where 

The characteristic equation is again a fourth-order polynomial  

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