Since the seminal work by A. Turing [1] in which he already studied a pattern on a sphere, many researchers have investigated the pattern formation on curved surfaces such as spheres and tori. Notably, all of these studies presumed that the Turing pattern remains static irrespective of the surface shape. Here, we show that the Turing pattern moves on curved surfaces in general [2]. Numerical and theoretical analyses reveal that there exist propagating solutions along the azimuth direction, and both the symmetries of the surface and pattern are related with onset of the pattern propagation. We also conduct weakly non-linear analysis and derive the amplitude equations, by which we show that the intricate interaction between modes arises on curved surfaces and results in pattern propagation and even more complex behaviors such as oscillatory and chaotic pattern dynamics. This study provides a novel and generic mechanism of pattern propagation that is caused by surface curvature.
[1] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237, 37-72 (1952).
[2] R. Nishide, and S. Ishihara, Pattern Propagation Driven by Surface Curvature, Phys. Rev. Lett. 128, 224101 (2022) / arXive:e2403.12442 / arXive:e2403.12444