The Existence of One Dimensional Steady Detonation Waves in a Simple Model Problem

We study a system of three nonlinear ordinary differential equations arising from one dimensional steady combustion problems. The four singular points in the phase space correspond to equilibrium solutions before and after burning has taken place, and the desired detonation wave is a trajectory connecting two of these points (the cold and hot boundary points). In this paper we show that the problem of finding steady detonations is equivalent to proving the existence and uniqueness of a transversal heteroclinic orbit. The presence of a large parameter, the activation energy, permits us to show analytically that such an orbit exists for an open set of parameter values. In doing so, we are able to give a geometric interpretation of the cold boundary difficulty. We conclude with some numerical results for lower activation energies.