Development of kinks in car-following models

Area of Science:

• Physics
• Applied Mathematics
• Traffic Engineering

Background:

• Car-following models in traffic flow theory often exhibit absolute stability.
• Near absolute stability, these models simplify to a modified Korteweg-deVries (mKdV) equation with minor corrections.
• Hyperbolic-tangent kink solutions of the mKdV equation represent transitions between different traffic spacings.

Purpose of the Study:

• To re-evaluate the selection mechanism of kink solutions in traffic flow models.
• To investigate the validity of assumptions in standard solvability analysis.
• To explore the full family of traveling wave solutions and their stability.

Main Methods:

• Analysis of modified Korteweg-deVries (mKdV) equation.
• Solvability analysis with critical examination of assumptions.
• Multiple-time-scales perturbation calculation.
• Identification and characterization of traveling wave solutions.

Main Results:

• The standard solvability analysis for mKdV kink solutions relies on an unjustified assumption.
• Without this assumption, the analysis yields a first-order correction to traffic spacing, not a selection criterion.
• A two-parameter family of traveling wave solutions, including kinks, was identified.
• Conditions were found for the decay, selection, or evolution of embedded traffic regions into kink pairs.

Conclusions:

• The presumed selection of a single kink solution in traffic flow models is an artifact of an unstated assumption.
• A richer set of traveling wave solutions exists for these traffic models.
• The dynamics of embedded traffic regions depend on specific conditions, leading to decay or transformation into kink pairs.