Rayleigh-Taylor instability of steady fronts described by the Kuramoto-Sivashinsky equation

We study steady thin reaction fronts described by the Kuramoto-Sivashinsky equation that separates fluids of different densities. This system may lead to hydrodynamic instabilities as buoyancy forces interact with the propagating fronts in a two-dimensional slab. We use Darcy's law to describe the fluid motion in this geometry. Steady front profiles can be flat, axisymmetric, or nonaxisymmetric, depending on the slab width, the density gradient, and fluid viscosity. Unstable flat fronts can be stabilized having a density gradient with the less dense fluid on top of a denser fluid. We find the steady front solutions from the nonlinear equations executing a linear stability analysis to determine their stability. We show regions of bistability where stable nonaxisymmetric and axisymmetric fronts can coexist. We also consider the stability of steady solutions in large domains, which can be constructed by dividing the domain into smaller parts or cells.