  In this paper, we propose the ultra-discrete optimal velocity model, a
cellular-automaton model for traffic flow, by applying the ultra-discrete
method for the optimal velocity model. The optimal velocity model, defined by a
differential equation, is one of the most important models; in particular, it
successfully reproduces the instability of high-flux traffic. It is often
pointed out that there is a close relation between the optimal velocity model
and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method
enables one to reduce soliton equations to cellular automata which inherit the
solitonic nature, such as an infinite number of conservation laws, and soliton
solutions. We find that the theory of soliton equations is available for
generic differential equations, and the simulation results reveal that the
model obtained reproduces both absolutely unstable and convectively unstable
flows as well as the optimal velocity model.
