Some exact solutions to the Lighthill-Whitham-Richards-Payne traffic flow equations (Q2854118)

The authors study the Lighthill-Whitham-Richards-Payne (LWRP) traffic flow equations given by the system of partial differential equations NEWLINE\[NEWLINE \rho_t + u \rho_x = - \rho u_x, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u_t + u u_x = (V(\rho)- u))/\tau_0 - (\nu_0/\rho) \rho_x. NEWLINE\]NEWLINE They introduce Lagrangian coordinates to eliminate certain nonlinearities.NEWLINENEWLINEFor exponential profiles of the initial fluid densities of the form NEWLINE\[NEWLINE \rho_0(x) = a \exp(\pm \lambda \xi), NEWLINE\]NEWLINE they find an explicit expression for the fluid density \(\rho\) in terms of \(X\) and \(t\). In this representation, the Lambert W-function appears.NEWLINENEWLINETo obtain the solution, the system is transformed to a nonlinear second-order equation that can be factorized as a product of two first-order operators similar to the wave equation.