Some exact solutions to the Lighthill Whitham Richards Payne traffic flow equations

TL;DR

This paper derives exact solutions to the Lighthill-Whitham-Richards-Payne traffic flow equations using transformations and special functions, revealing traffic behavior akin to soliton solutions and potentially applicable to other fluid dynamics problems.

Contribution

It introduces a novel method for obtaining exact solutions to the LWRP equations through consecutive lagrangian transformations and Lambert functions, providing insights into traffic flow dynamics.

Findings

Exact formulas for density and velocity depending on initial conditions

Traffic lineup often splits into two uniform velocity streams

Solutions resemble soliton behavior in fluid dynamics

Abstract

We find a class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. Using two consecutive lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulas for the dependence of the car density and velocity on x and t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg-de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.