Laminarizing effects of dispersion in an active-dissipative nonlinear medium

Abstract

The Kuramoto-Sivashinsky equation has become a popular prototype for systems which exhibit spatial-temporal chaos. We show here that a linear dispersion term δ h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures, as is consistent with prior numerical and experimental observations. The study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation. It is shown that by δ > 1.1, the infinite families of stationary periodic waves of the Kuramoto-Sivashinsky equation, each ending in a solitary wave, have been annihilated successively such that only a lone family of periodic waves, consisting of one-hump KdV pulses for δ > 3.7, remains as the only periodic stationary wave attractors of the system. These periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial-temporal chaos in an extended domain with significant dispersion.