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The detonation wave stability is addressed using Fickett's equation, i.e.,the reactive form of Burgers' equation. This serves as a simple analogue to thereactive Euler equations, permitting one to gain insight into the nonlineardynamics of detonation waves. Chemical kinetics were modeled using a two-stepreaction with distinct induction and reaction zones. An evolution equation forthe detonation structure was derived using the method of matched asymptoticsfor large activation energy and slow rate of energy release. While the firstorder solution was found unconditionally unstable, the second order evolutionequation predicted both stable and unstable solutions. The neutral stabilityboundary was found analytically, given by $\chi=4$, where $\chi$ is the productof activation energy and the ratio of induction to reaction time. Thisreproduces accurately what has been previously established for the reactiveEuler equations and verified experimentally. The evolution equation alsocaptures stable limit cycle oscillations in the unstable regime and offersunique insight into the instability mechanism. The mechanism amplifying theperturbations lies within the induction zone, where the Arrhenius-type rateequation provides a large change in induction times for small perturbations.The mechanism attenuating the perturbations arises from acoustic effects, whichdelays the amplification of the shock front. The longer the detonation wave,the more time it takes for the amplification from the reaction zone to reachthe shock front, creating gradients that counter-act the amplification from theflame acceleration. The results agree with direct numerical simulation, as wellas recovering many similarities with the reactive Euler equation.

A nonlinear evolution equation for pulsating detonations using Fickett's model with chain branching kinetics