  Continuing the program initiated by Humpherys, Lyng, & Zumbrun [17] for
strong detonation waves, we use a combination of analytical and numerical
Evans-function techniques to analyze the spectral stability of weak detonation
waves in a simplified model for gas-dynamical combustion. Combining these new
spectral stability results with the pointwise Green function analysis of Lyng,
Raoofi, Texier, & Zumbrun [22], we conclude that these waves are nonlinearly
stable. The principal novelty of this analysis is the treatment of weak
detonation waves. In contrast to the case of strong detonation waves, weak
detonation waves are undercompressive and the stability of these waves is
delicate and has not been treated by standard weighted-norm techniques. The
present analysis thus provides a case study illustrating the flexibility and
power of the Evans-function- based approach to stability. As in the case of
strong detonations, we find that all tested waves are spectrally stable, hence
nonlinearly stable.
