Some exact solutions to the Lighthill–Whitham–Richards–Payne traffic flow equations: II. Moderate congestion

DOI10.1088/1751-8113/47/41/415203zbMATH Open1308.35247arXiv1310.6946OpenAlexW2076289050MaRDI QIDQ2927696

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Publication date: 4 November 2014

Published in: Journal of Physics A: Mathematical and Theoretical (Search for Journal in Brave)

Abstract: We find a further class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either obtain exact formulae for the dependence of the car density and velocity on x, 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 not only Rowlands, Infeld and Skorupski J. Phys. A: Math. Theor. 46 (2013) 365202 (part I) but also the two soliton solution to the Korteweg-de Vries equation. This paper can be read independently of part I. This explains unavoidable repetitions. Possible uses of both papers in checking numerical codes are indicated at the end. Since LWRP, numerous more elaborate models, including multiple lanes, traffic jams, tollgates etc. abound in the literature. However, we present an exact solution. These are few and far between, other then found by inverse scattering. The literature for various models, including ours, is given. The methods used here and in part I may be useful in solving other problems, such as shallow water flow.

Full work available at URL: https://arxiv.org/abs/1310.6946

zbMATH Keywords

Korteweg-de Vries equation Lagrangian coordinates traffic models exact solutions of partial differential equations

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Traffic problems in operations research (90B20) Soliton equations (35Q51) Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems (37K40) PDEs in connection with mathematical programming (35Q90)

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