On the pattern formation mechanism of (2 + 1)D max-plus models

We propose a max-plus equation which reproduces evolutional patterns often observed in reaction-diffusion systems of excitable media. The equation gives a travelling wave, a target pattern and a spiral pattern from appropriate initial data. Moreover, using the advantages of max-plus equations, we obtain the solutions exactly by a reduction from the high-dimensional equation to a lower one. In the reduction, we use coordinate curves according to a pattern shape. It is interesting that all patterns satisfy the same reduced equation. We also propose two other models similar to the previous one and discuss the behaviour of solutions.