Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii Equation

Abstract

We prove the existence of a family of travelling wave solutions in a variant of the $\text{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, 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 provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using $\text{high activation energy asymptotics}$. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function $\overline{c}(\epsilon )$, where $0<\epsilon \ll 1$ is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of travelling wave solutions for non-minimal wave speeds $c>\overline{c}(\epsilon )$. The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as $\epsilon \to 0$, which we refer to herein as the $\text{convective-diffusive}$ and $\text{diffusive-reactive}$ zones. The main idea of the proof is to use the geometric blow-up method to identify and characterise a $(c,\epsilon )$-family of heteroclinic orbits which traverse both of these regimes, and correspond to travelling 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 dynamical systems with singular exponential nonlinearities.

arXiv commentAccepted version, to appear in SIADS