# Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation

### Gregory I. Sivashinsky

Tel-Aviv University, Israel### Michael L. Frankel

Indiana University Purdue University Indianapolis, USA### Josephus Hulshof

Vrije Universiteit, Amsterdam, Netherlands### Claude-Michel Brauner

Université de Bordeaux I, Talence, France

## Abstract

We consider a quasi-steady version of the $\kappa-\theta$ model of flame front dynamics introduced in \cite{FGS03} . In this case the mathematical model reduces to a single integro-differential equation. We show that a periodic problem for the latter equation is globally well-posed in Sobolev spaces of periodic functions. We prove that near the instability threshold the solutions of the equation are arbitrarily close to these of the Kuramoto-Sivashinsky equation on a fixed time interval if the evolution starts from close configurations. We present numerical simulations that illustrate the theoretical results, and also demonstrate the ability of the quasi-steady equation to generate chaotic cellular dynamics.