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

TL;DR

This paper proves the existence and properties of travelling wave solutions in a reaction-diffusion model with exponential nonlinearities, using geometric blow-up methods to analyze complex asymptotic regimes relevant to combustion theory.

Contribution

It provides a rigorous analysis of travelling waves in the Zeldovich-Frank-Kamenetskii equation, including smoothness of wave speed and asymptotic series for slow manifolds, extending previous formal asymptotics.

Findings

Existence of a family of travelling wave solutions in the ZFK equation.

Proof of smoothness of the minimum wave speed function.

Asymptotic series for the slow manifold in the high activation energy limit.

Abstract

We prove the existence of a family of travelling wave solutions in a variant of the $\textit{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, 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 $\textit{high activation energy asymptotics}$. 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 $\textit{convective-diffusive}$ and $\textit{diffusive-reactive}$ 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.