KdV cnoidal waves in a traffic flow model with periodic boundaries

TL;DR

This paper applies modulation theory to a perturbed KdV equation derived from an optimal-velocity traffic model, revealing periodic wave solutions that describe vehicle headway and confirming findings with numerical simulations.

Contribution

It introduces a modulation theory approach to analyze cnoidal wave solutions in a traffic flow model, extending previous soliton analysis to periodic waves with variable modulation.

Findings

Derived Whitham equations for the perturbed KdV model.

Identified a family of spatially periodic cnoidal wave solutions.

Established the relationship between wave speed and modulation parameters.

Abstract

An optimal-velocity (OV) model describes car motion on a single lane road. In particular, near to the boundary signifying the onset of traffic jams, this model reduces to a perturbed Korteweg-de Vries (KdV) equation using asymptotic analysis. Previously, the KdV soliton solution has then been found and compared to numerical results (see Muramatsu and Nagatani (1999)). Here, we instead apply modulation theory to this perturbed KdV equation to obtain at leading order, the modulated cnoidal wave solution. At the next order, the Whitham equations are derived, which have been modified due to the equation perturbation terms. Next, from this modulation system, a family of spatially periodic cnoidal waves are identified that characterise vehicle headway distance. Then, for this set of solutions, we establish the relationship between the wave speed and the modulation term, which is dependent upon the number of oscillations over the spatial domain. This analysis is confirmed with comparisons to numerical solutions of the OV model.