Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts

Abstract

The Sivashinsky integral equation governing certain hydrodynamical instabilities of one-dimensional flame fronts is a special case of Lee and Chen's (Phys. Scr. 2 (1982) 41) non linear plasma models; as such it has a pole decomposition. This explains the highly organized structures observed in numerical simulations. The Sivashinsky equation has stable steady solutions with the poles aligned parallel to the imaginary axis. With periodic boundary conditions, when the number of linearly unstable modes is large, the poles condense into a In coth distribution. This is illustrated by numerical calculations of equilibrium positions of poles. The pole condensation explains cusp-like wrinkles in certain flame fronts. The energy spectrum for the front displacement follows a In2 k law.