Stability of pole solutions for planar propagating flames

It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions. For such solutions, the original PDE can be reduced to a set of ordinary differential equations (ODE's). The situation, however, is paradoxical since the steady solutions obtained by numerically integrating the PDE differ, in general, front the exact solutions governed by the ODE's. For example, if the initial condition is a one-pole steady solution, provided that the size of the domain considered is larger than a (small) critical length, the number of poles increases with time in the PDE while it remains constant in the ODE's. In previous studies, this generation of poles was thus believed to be an artifact or product of external noise, rather than a dynamical process intrinsic to the PDE. In this paper, we show that the phenomenon is due to the fact that most exact steady pole solutions are unstable for the PDE. In certain cases, such solutions are unstable for the ODE's, in other cases, they are neutrally stable for the ODE's but unstable for the PDE. The only steady pole solutions which are neutrally stable for both the ODE's and the PDE correspond to small interval lengths; both their number of poles and propagation speed are maximal (among all possible steady solutions corresponding to the interval considered) and all their poles are aligned on the same vertical axis in the complex plane (i.e., such solutions are coalescent). For a given interval of small length, there is only one such solution (up to translation symmetry).