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We introduce discrete and -adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional -adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the -adic unit ball. [ABSTRACT FROM AUTHOR]

We introduce a new family of -adic nonlinear evolution equations. We establish the local well-posedness of the Cauchy problem for these equations in Sobolev-type spaces. For a certain subfamily, we show that the blow-up phenomenon occurs and provide numerical simulations showing this phenomenon. [ABSTRACT FROM AUTHOR]