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

例如，在自传播高温合成（SHS）中观察到螺旋波。当燃烧波保持其轮廓并以恒定速度传播时，可以通过SHS产生高质量的均匀产品。然而，当实验条件改变时，平面行波可能会失去稳定性，产生一些不均匀的行波。实际上，平面脉动波是通过平面行波的Hopf分岔产生的。此外，我们观察到以螺旋形式传播的波围绕具有几个反应点的圆柱形样品。这种波被称为螺旋波，因为我们的三维数值模拟显示，波的等温表面有一些翅膀，随着时间的推移，它呈螺旋状向下旋转。在实验室聚合的传播前沿也观察到类似的螺旋波，并通过一些自催化反应和SHS的数值模拟得到了类似的螺旋波。我们研究了螺旋波的存在条件和波型从行模到脉动模和/或螺旋模的转变过程，得到了以下结果：稳定的螺旋波可以从平面行波中直接分岔出来。即使行波在R内是稳定的，其对应的平面行波在带域和圆柱域都可能是不稳定的，而螺旋波代替了平面行波。当带宽L较小时或圆柱畴半径R较小时，不存在稳定的螺旋波。不同反应点数的螺旋波可以稳定共存。

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.