Non-linear wave dynamics and stability transitions in traffic flow with localised market perturbations

Urban traffic in sub-Saharan Africa exhibits strong interactions between formal vehicular flow and informal roadside activities. This paper develops a macroscopic second-order traffic flow model in which market-induced disturbances are represented as spatio-temporally localised perturbations to equilibrium velocity, anticipation, and pedestrian impedance. By embedding these effects in a viscous continuum framework, we show how informal markets fundamentally alter the collective dynamics of vehicular flow. Linear stability analysis demonstrates that market intensity raises the critical stability threshold, narrowing the domain of stable operation. A non-linear multiple-scale expansion reduces the governing equations to a Korteweg–de Vries-type system, revealing the emergence of solitary waves as organising structures near criticality. Numerical simulations across free-flow, moderate-flow, and near-congested regimes confirm that localised perturbations can evolve into oscillatory waves or stationary jam fronts, with pronounced hysteresis effects at high density. These results demonstrate that roadside markets act as destabilising agents in traffic systems by inducing phase-transition-like shifts between free flow, oscillatory states, and congestion. The findings illustrate how localised human activity modifies non-linear flow dynamics, providing a broader perspective on the role of spatio-temporal heterogeneities in complex transport systems.