Mathematical Analysis of helical waves arising in some-reaction diffusion systems

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A helical wave is observed in self-propagating high-temperature syntheses (SHS), for instance. One can create a high-quality uniform product by the SHS when a combustion wave keeps its profile and propagates at a constant velocity. When experimental conditions are changed, however, the planar traveling wave may lose its stability and give place some non-uniform ones. Actually, a planar pulsating wave appears through the Hopf bifurcation of planar traveling wave. Moreover, we observe a wave that propagates in the form of spiral encircling the cylindrical sample with several reaction spots. This wave is called a helical wave since it has been shown by our 3D numerical simulation that the isothermal surface of the wave has some wings and it helically rotates down as time passes on. Similar helical waves are observed also in propagation fronts of polymerizations in laboratory and they are obtained also by numerical simulation of some autocatalytic reactions as well as the SHS.We have been studied the existing condition of helical wave and the transition process of wave patterns from traveling mode to pulsating mode and/or helical mode, and we have obtained the following results:1. A stable helical wave can bifurcate directly from a planar traveling wave.2. Even if a traveling wave is stable in R, the corresponding planar traveling wave can be unstable in the band domain as well as in the cylindrical domain, and a helical wave takes the place of planar traveling wave.3. There are no stable helical wave when the band width L is small or the radius R of cylindrical domain is small.4. Helical waves with different numbers of reaction spots can coexist stably.