ON A SYSTEM OF NONLINEAR HYPERBOLIC CONSERVATION-LAWS WITH SOURCES

We study the existence of travelling reaction fronts connecting equilibrium states in the nonlinear reactive-convective system u(t) + (1/2u2 + qlambda)x = (2-u)(u-1) lambda(t) = r(u,lambda), where r represents a given chemical reaction rate of a reversible reaction A half arrow right over half arrow left B, and f(u) = (2-u) (u-1) represents a nonlinear heat source. The original model equations for irreversible reactions and without sources were motivated by an analog of reactive flow and detonation processes posed by Fickett and Majda. It is shown that endothermic compressions exist for wave speeds c greater-than-or-equal-to 2 and in this case, a singular perturbation method is developed to find a two-term analytic approximation for the waveforms. For wavespeeds 0 < c < c* < 1, for some c* = c*(q), nonmonotonic travelling waves exist. These subsonic travelling waves have either an oscillating tail or a minimum occurring at a finite value, and they essentially represent exothermic rarefaction waves. In this latter case a singular line in the flow comes into play, and a method of analysis is developed based on desingularization of the phase plane equations.