Traveling and spiral waves for sequential flames with translation symmetry: coupled CGL–Burgers equations

The dynamics of traveling and spiral waves resulting from the oscillatory instability of a uniformly propagating planar flame front governed by a sequential reaction is investigated. The nonlinear dynamics of this system, which enjoys translation symmetry, is described by two coupled equations: a complex Ginzburg–Landau equation for the oscillation amplitude, and a Burgers equation for the slow Goldstone mode due to the symmetry, describing frontal deformation. The pure CGL equation in 2D exhibits spiral wave solutions as well as uniformly propagating wave solutions. We present a stability analysis and numerical computations of the coupled system to show how the coupling modifies traveling and spiral wave solutions of the pure CGL equation. In particular, we show that the coupled system exhibits new types of instabilities as well as new dynamical behavior, including bound states of two or four spirals, “liquid spiral” states, superspiral structures, oscillating cellular structures separated by chaotically merging and splitting domain walls, and others.