Traveling Waves and Exponential Nonlinearities in the Zeldovich–Frank-Kamenetskii Equation

We prove the existence of a family of traveling wave solutions in a variant of the Zeldovich–Frank-Kamenetskii (ZFK) equation, a reaction-diffusion equation which models the propagation of planar laminar premixed flames in combustion theory. Our results are valid in an asymptotic regime which corresponds to a reaction with high activation energy, and they provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using high activation energy asymptotics. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function c̄(ε), 0< ε<<1 where is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of traveling wave solutions for nonminimal wave speeds c> c̄(ε). The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as ε→0, which we refer to herein as the convective-diffusive and diffusive-reactive zones. The main idea of the proof is to use the geometric blow-up method to identify and characterize a (c, ε)-family of heteroclinic orbits which traverse both of these regimes, and correspond to traveling waves in the original ZFK equation. More generally, our analysis contributes to a growing number of studies which demonstrate the utility of geometric blow-up approaches to the study of dynamical systems with singular exponential nonlinearities.