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In this paper, we show that a variant of the discrete Burgers equation can beobtained through the Cole--Hopf transformation to a generalized discretediffusion equation corresponding to the correlated random walk, which is alsoknown as a generalization of the well known random walk. By applying thetechnique called ultradiscretization, we obtain the generalized ultradiscretediffusion equation, the ultradiscrete Cole--Hopf transformation and a variantof the ultradiscrete Burgers equation. Moreover, we show that the resultingultradiscrete Burgers equation yields cellular automata which can beinterpreted as a traffic flow model.

A variant of the discrete Burgers equation derived from the correlated random walk and its ultradiscretization