Asymptotic theory of traffic jams

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Abstract
Based on singular perturbation methods, an asymptotic theory of traffic jams of large amplitude is developed. Simple equations describing the form of traffic jams of large amplitude are found. The theory leads to analytical formulas for the characteristic, i.e., intrinsic or unique, parameters of traffic flow (such as the average velocity of the downstream front of a wide jam, as well as the flux, density and average vehicle speed of the outflow from the jam) which are independent of the road length, the vehicle density of the initial traffic flow, or other initial conditions. Analytical investigations have been made that show that, in agreement with earlier numerical results [B. S. Kerner and P. Konhäuser, Phys. Rev. E 50, 54 (1994)], the boundary (threshold) flux at which a traffic jam can still exist is equal to the flux in the outflow from a jam. The manner in which the shape of a traffic jam evolves due to changes in the initial vehicle density is analytically investigated. Simple analytical formulas are obtained for the parameters of narrow traffic jams capable of forming in a limited interval of vehicle densities. A comparison is also made between the results of the present analytical theory of traffic jams, the theory of shock waves in gas dynamics, the classical Lighthill-Whitham-theory [M. J. Lighthill and B. G. Whitham, Proc. R. Soc. London Ser. A 229, 317 (1955)] of kinematic waves, and the recently discovered experimental features and characteristics of wide traffic jams in actual traffic.