Bifurcations in a planar propagating flame as the size of the domain increases

The partial differential equation (PDE) describing the dynamics of hydrodyn amically unstable planar flame front has exact pole solutions which satisfy a set of ordinary differential equations (ODEs). This set of ODEs prohibit s the creation of new poles in the complex plane, or the appearance of cusp s in the physical space, as observed experimentally. The contribution of th is paper is to show that most exact pole solutions are unstable solutions f or the PDE. Even the one-peak, coalescent solutions (whose number of poles is maximal) is unstable as soon as the number of poles exceeds a certain (r ather small) threshold. As the size of the domain increases, the front unde rgoes bifurcations which can be described as follows: the one-pole, one-pea k coalescent solution is neutrally stable for small intervals. As the lengt h of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is neutrally stable, For larger intervals, the two-pole solution is unstable, the three-pole solution becomes stable. As the inter val length increases further, the steady one-peak, coalescent solutions are no longer stable and bifurcations to unsteady states occur. (C) 1999 Elsev ier Science B.V. and IMACS, All rights reserved.