Concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique

Following the proposal of a filtration technique by Nobe, Satsuma and Tokihiro, we concretely construct partial difference equations, which preserve any time evolution patterns of cellular automaton (CA) stably by the filtration technique. We illustrate how to develop a method of filtration for applying to the typical two spatial dimensional CA rule—the game of life—and verify that the filtration method provides the stable difference equation associated with the CA, compared with the inverse ultradiscretization. Besides, in order to discuss whether the filtration technique can lead one to partial differential equations from CA rules, we show a derivation of the Burgers equation from Rule 184 CA via the discrete Burgers equation constructed by the filtration method as an example.