Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model

Abstract

A multiple-scale analysis of the Kuramoto-Sivashinsky one-dimensional model of a flame front with 2π-periodic boundary conditions is presented. For arbitrary large values of the number M of linearly unstable modes there exist stable steady solutions of period 2π/N where N = O(M). These ‘cellular solutions’ exhibit elastic behaviour under perturbations of wavelength much larger than 2π/N. The results are illustrated by numerical experiments. Elasticity has its origin in the translation and Galilean invariances. Similar invariance properties are likely to be at the root of the viscoelastic behaviour of turbulent flows conjectured by many authors.