Abstract
Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius $R$ and minor radius $r$. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on the higher-order matched asymptotic expansion with the analytic expression of the Green's function of the Laplace--Beltrami operator on the toroidal surface. Owing to the analytic representation, one can investigate the existence of equilibria with a single spot, two spots, and the ring configuration where $N$ localized spots are equally spaced along a latitudinal line with mathematical rigor. We show that localized spots at the innermost/outermost locations of the torus are equilibria for any aspect ratio $\alpha=\frac{R}{r}$. In addition, we find that there exists a range of the aspect ratio in which localized spots stay at a special location of the torus. The theoretical results and the linear stability of these spot equilibria are confirmed by solving the nonlinear evolution of the Brusselator reaction-diffusion model by numerical means. We also compare the spot dynamics with the point vortex dynamics, which is another model of spot structures.
Highlights
Self-organizing beautiful patterns of localized spot-like structures appear ubiquitously in many natural phenomena
A regular/irregular lattice of spot structures is formed in Bose--Einstein condensates (BECs) [1, 13]
It is experimentally observed that such localized spot patterns emerge in a ferrocyanide-iodate-sulphite reaction [18], a chlorine dioxide-iodine-malonic acid reaction [10], and a gas charge system [3, 4]
Summary
Self-organizing beautiful patterns of localized spot-like structures appear ubiquitously in many natural phenomena. Owing to the geophysical and biological relevance, there are many studies on pattern formations of localized spot structures on the surface of a sphere It is shown in [21] that point vortices become a vortex crystal when they are placed on the vertices of regular polyhedrons, and the relation between the configurations and the optimal packing problem is discussed. Towards the applications to superfluids, the evolution equation of vortex dynamics on the toroidal surface has been derived in [28], in which some vortex crystals are constructed, and the dynamics of one and two point vortices are investigated It has been shown in [29] that the stability of a ring configuration of N point vortices changes depending on the sign of curvature and the modulus \alpha. They numerically investigate the stability of one and two localized spots
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