Bifurcation of a helical wave from a traveling wave

Spin mode instabilities have been observed in solid-state combustion reactions and in propagating fronts of polymerization among others, that is, when some conditions are changed, a planar traveling wave may lose its stability and there appears a planar wave or a helical wave propagating in the form of spiral encircling the cylindrical sample with several reaction spots. In the present paper, we study the existing condition of stable helical waves and the transition process of wave patterns from a static mode to pulsating and/or helical modes. By using a full system of reaction-diffusion equations, we clarify that a stable helical wave can bifurcate directly from a planar traveling wave and that helical waves with different numbers of reaction spots can coexist stably.