Travelling waves in exothermic systems

The classical theoretical problem of thermal ignition and extinction in a reactive slab of infinite extent under conditions near transition to continuous behaviour is revisited. It is assumed that the system is governed by two parameters. The first corresponds to the Frank-Kamenetskii parameter, $\delta$; the second is in some circumstances related to the dimensionless ambient temperature of inverse activation energy ($\beta = RT \_a/E$) and in other circumstances to the dimensionless adiabatic temperature rise ($\theta\_{ad}$ or B). The value of the second parameter ($\beta$ or B) is assumed to be close to its transition value, where a `cuspoidal' behaviour of the reacting system appears. A perturbation analysis of the problem shows that additional, spatially distributed states exist in the system in the form of travelling waves of reaction. One of the newly discovered solutions is stable and corresponds to the one-dimensional combustion wave. The second solution is unstable and cannot be related to a real physical situation.