  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 the
reactive Euler equations, permitting one to gain insight into the nonlinear
dynamics of detonation waves. Chemical kinetics were modeled using a two-step
reaction with distinct induction and reaction zones. An evolution equation for
the detonation structure was derived using the method of matched asymptotics
for large activation energy and slow rate of energy release. While the first
order solution was found unconditionally unstable, the second order evolution
equation predicted both stable and unstable solutions. The neutral stability
boundary was found analytically, given by $\chi=4$, where $\chi$ is the product
of activation energy and the ratio of induction to reaction time. This
reproduces accurately what has been previously established for the reactive
Euler equations and verified experimentally. The evolution equation also
captures stable limit cycle oscillations in the unstable regime and offers
unique insight into the instability mechanism. The mechanism amplifying the
perturbations lies within the induction zone, where the Arrhenius-type rate
equation provides a large change in induction times for small perturbations.
The mechanism attenuating the perturbations arises from acoustic effects, which
delays the amplification of the shock front. The longer the detonation wave,
the more time it takes for the amplification from the reaction zone to reach
the shock front, creating gradients that counter-act the amplification from the
flame acceleration. The results agree with direct numerical simulation, as well
as recovering many similarities with the reactive Euler equation.
