In this paper we introduce a simple combustion model and study its properties. The model supports both deflagration and detonation waves and exhibits instability. Our purpose is to introduce an admissibility criterion through the study of elementary waves, and use it to investigate the nonlinear stability and instability of flows with combustion waves. In particular, our model exhibits instability for unburnt states if the binding energy is sufficiently large. Because reactive gas flow is highly unstable, this aspect of our model is physically reasonable. The simplicity of our model allows us to make a fairly general study of solutions. A scalar value u is to represent a lumped quantity of the gas flow such as density, velocity or temperature, q, which denotes the binding energy of the reactive gas, equals a constant qo for unburnt gas and zero for burnt gas. A typical point in Lagrangian coordinates is denoted by x E R 1. In our model,