Propagation of wave modes and antispiral waves in a reaction-diffusion system

Since the observation of antispirals by Vanag et al. in a reaction-diffusion system, the study of its origin becomes a focus of nonlinear science and pattern formation. It was shown that antispirals may exist in a reaction-diffusion system when the system is near the onset of Hopf bifurcation. Here, we demonstrate that antispiral waves can also exist in a relaxational oscillatory medium, which is far from Hopf onset. Furthermore, we clarify the previously unclear concept of group velocity in inharmonic nonlinear waves, and prove that the group velocity v(g)=domega/dk is physically significant and valid both for excitable media and for oscillatory media. From this formula, we identify two types of wave propagating fronts: gentle mode front and steep mode front. Finally, we discuss the origin of counterpropagating waves and propose necessary conditions for antispiral formation. A direct deduction from these criteria is that antispirals cannot exist in an excitable medium.