A complex Swift–Hohenberg equation coupled to the Goldstone mode in the nonlinear dynamics of flames

The nonlinear dynamics of a propagating flame front governed by a two-stage sequential chemical reaction is considered in the parameter range where the uniformly propagating front is unstable. We show that near the transition from the short wave to the long wave oscillatory instability the nonlinear dynamics is described by a Swift–Hohenberg equation with dominant dispersive term, coupled to an evolution equation for the zero mode associated with the translation symmetry of the propagating wave. The nonlinear dynamics described by this system of equations is studied both analytically and numerically. In the case of weak coupling between the two equations, we observe the spontaneous formation of spiral waves with rapidly moving cores, while strong coupling leads either to chaotic dynamics or to the formation of oscillons—spatially localized oscillating structures.