Pattern-formation characteristics of interacting kinematic waves

Abstract

This work presents a model for the formation of density patterns in certain one-dimensional systems where the particle flux is a well-defined function of particle density (e.g., traffic flow or granular flow in a tube). In these systems, macroscopic regions characterized by large density contrasts are observed to evolve from very small-scale fluctuations. This paper shows that such patterns develop naturally when the small-scale noise is viewed as a set of stable density regions which propagate according to the formalism of kinematic waves. In the theory of kinematic waves, the interface separating a region of density ${\mathrm{\rho }}_1$ from a region of density ${\mathrm{\rho }}_2$ moves with a velocity v=[j(${\mathrm{\rho }}_1$)-j(${\mathrm{\rho }}_2$)]/(${\mathrm{\rho }}_1$-${\mathrm{\rho }}_2$), where j(ρ) is the flux at density ρ. With interfaces propagating according to this equation, both analytic and numerical results indicate that the noisy state is quickly replaced by a state in which neighboring density regions have a very large density contrast. Thus interacting kinematic waves and small-scale fluctuations are all that is necessary for this pattern formation.