Phase transition in a difference equation model of traffic flow

Development of kinks in car-following models

Many car-following models of traffic flow admit the possibility of absolute stability, a situation in which uniform traffic flow at any spacing is linearly stable. Near the threshold of absolute stability, these models can often be reduced to a modified Korteweg-deVries (mKdV) equation plus small corrections. The hyperbolic-tangent "kink" solutions of the mKdV equation are usually of particular interest, as they represent transition zones between regions of different traffic spacings. Solvability analysis is believed to show that only a single member of the one-parameter family of kink solutions is preserved by the correction terms, and this is interpreted as a kind of selection. We show, however, that the usual solvability calculation rests on an unstated, unjustified assumption, and that without this assumption it merely gives a first-order correction to the relation between the traffic spacings far behind and far ahead of the kink, rather than any kind of "selection" criterion for the family of kink solutions. On the other hand, we display a two-parameter family of traveling wave solutions of the mKdV equation, which describe regions of one traffic spacing embedded in traffic of a different spacing; this family includes the kink solutions as a limiting case. We carry out a multiple-time-scales calculation and find conditions under which the inclusions decay, conditions that lead to a selected inclusion, and conditions for which the inclusion evolves into a pair of kinks.

Solitons and kinks in a general car-following model

We study a general car-following model of traffic flow on an infinitely long single-lane road, which assumes that a car's acceleration depends on time-delayed values of its own speed, the headway between it and the car ahead, and the rate of change of headway, but makes minimal assumptions about the functional form of that dependence. We present a detailed characterization of the onset of linear instability; in particular we find a specific limit on the delay time below which the marginal wave number at the onset of instability is zero, and another specific limit on the delay time above which steady flow is always unstable. Crucially, the threshold of absolute stability generally does not coincide with an inflection point of the steady-state velocity function. When the marginal perturbation at onset has wave number 0, we show that Burgers and Korteweg-de Vries (KdV) equations can be derived under the usual assumptions, and that corrections to the KdV equation "select" a single member of the one-parameter set of its one-soliton solutions by driving a slow evolution of the soliton parameter. While in previous models this selected soliton has always marked the threshold of a finite-amplitude instability of linearly stable steady flow, we find that it can alternatively be a stable, small-amplitude jam that occurs when steady flow is linearly unstable. The model reduces to the usual modified Korteweg-de Vries (mKdV) equation only in the special situation that the threshold of absolute stability coincides with an inflection point of the steady-state velocity function; in general, near the threshold of absolute stability the model reduces instead to a KdV equation in the regime of small solitons, while near an inflection point it reduces to a Hayakawa-Nakanishi equation. Like the mKdV equation, the Hayakawa-Nakanishi equation admits a continuous family of kink solutions, and the selection criterion arising from the corrections to this equation can be written down explicitly.

Bunching transition in a time-headway model of a bus route

A time-headway model is presented to mimic bus behavior on the bus route. The motion of a bus is described in terms of the time headway between its bus and the bus in front. We study the bunching behavior of buses induced by interacting with other buses and passengers. It is shown that the dynamical phase transitions among the inhomogeneous bunching phase, the homogeneous free phase, the coexisting phase, and the homogeneous congested phase occur with varying the initial time headway. We study the effect of not stopping at bus stops on the time-headway profile. It is found that the bunching transition lines are consistent with the neutral stability curves obtained by the linear stability analysis.

Density waves in traffic flow

Density waves are investigated in the car-following model analytically and numerically. This work is a continuation of our previous investigation of traffic flow in the metastable and unstable regions [Phys. Rev. E 58, 4271 (1998); 60, 180 (1999)]. The Burgers equation is derived for the density wave in the stable region of traffic flow by the use of nonlinear analysis. It is shown, numerically, that the triangular shock wave appears as the density wave at the late stage in the stable region. The decay rate of the shock wave is calculated and compared with the analytical result. It is shown that the density waves out of the coexisting curve, near the spinodal line, and within the spinodal line appear, respectively, as the triangular shock wave, the soliton, and the kink-antikink wave. The density waves are described, respectively, by the Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations.

Traffic jams induced by fluctuation of a leading car

We present a phase diagram of the different kinds of congested traffic triggered by fluctuation of a leading car in an open system without sources and sinks. Traffic states and density waves are investigated numerically by varying the amplitude of fluctuation using a car following model. The phase transitions among the free traffic, oscillatory congested traffic, and homogeneous congested traffic occur by fluctuation of a leading car. With increasing the amplitude of fluctuation, the transition between the free traffic and oscillatory traffic occurs at lower density and the transition between the homogeneous congested traffic and the oscillatory traffic occurs at higher density. The oscillatory congested traffic corresponds to the coexisting phase. Also, the moving localized clusters appear just above the transition lines.

Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction

The car-following model of traffic is extended to take into account the car interaction before the next car ahead (the next-nearest-neighbor interaction). The traffic behavior of the extended car-following model is investigated numerically and analytically. It is shown that the next-nearest-neighbor interaction stabilizes the traffic flow. The jamming transition between the freely moving and jammed phases occurs at a higher density than the threshold of the original car-following model. By increasing the maximal velocity, the traffic current is enhanced without jam by the stabilization effect. The jamming transition is analyzed with the use of the linear stability and nonlinear perturbation methods. The traffic jam is described by the kink solution of the modified Korteweg-de Vries equation. The theoretical coexisting curve is in good agreement with the simulation result.

Chaotic jam and phase transition in traffic flow with passing

The lattice hydrodynamic model is presented to take into account the passing effect in one-dimensional traffic flow. When the passing constant gamma is small, the conventional jamming transition occurs between the uniform traffic and kink density wave flows. When passing constant gamma is larger than the critical value, the jamming transitions occur from the uniform traffic flow, through the chaotic density wave flow, to the kink density wave flow, with an increasing delay time. The chaotic region increases with passing constant gamma. The neutral stability line is derived from the linear stability analysis. The neutral stability line coincides with the transition line between the uniform traffic and density wave flows. The modified Korteweg-de Vries equation describing the kink jam is derived for small values of gamma by use of a nonlinear analysis.

Soliton and kink jams in traffic flow with open boundaries

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.