Travelling wave solutions of the perturbed mKdV equation that represent traffic congestion

TL;DR

This paper derives and analyzes steady travelling wave solutions of a perturbed mKdV equation modeling traffic flow, identifying conditions for congestion waves and validating results with numerical simulations.

Contribution

It introduces a multi-scale perturbation approach to find explicit travelling wave solutions in a traffic model, linking wave properties to driver sensitivity.

Findings

Existence of steady waves with constant amplitude and period

Identification of solutions corresponding to traffic congestion

Validation of analytical solutions through numerical simulations

Abstract

A well-known optimal velocity (OV) model describes vehicle motion along a single lane road, which reduces to a perturbed modified Korteweg-de Vries (mKdV) equation within the unstable regime. Steady travelling wave solutions to this equation are then derived with a multi-scale perturbation technique. The first order solution in the hierarchy is written in terms of slow and fast variables. At the following order, a system of differential equations are highlighted that govern the slowly evolving properties of the leading solution. Then, it is shown that the critical points of this system signify travelling waves without slow variation. As a result, a family of steady waves with constant amplitude and period are identified. When periodic boundary conditions are satisfied, these solutions' parameters are associated with the driver's sensitivity, $\hat{a}$, which appears in the OV model. For some given $\hat{a}$, solutions of both an upward and downward form exist, with the downward type corresponding to traffic congestion. Numerical simulations are used to validate the asymptotic analysis and also to examine the long-time behaviour of our solutions.