Generalized discrete and ultradiscrete Burgers equations derived through the correlated random walk

Abstract

The correlated random walk is known as a generalization of the well-known random walk. In this study, we present that a generalized discrete Burgers equation corresponding to the correlated random walk can be obtained through a Cole–Hopf transformation to a generalized discrete diffusion equation. By applying a technique called ultradiscretization, the generalized ultradiscrete diffusion equation, the ultradiscrete Cole–Hopf transformation, and a variant of the ultradiscrete Burgers equation are obtained. Additionally, this study shows that the resulting ultradiscrete Burgers equation yields cellular automata that can be interpreted as a traffic flow model with controllability of vehicle flow.