Singularities in kinematic wave theory: Solution properties, extended methods and duality revisited

According to Euler–Lagrange duality principle of kinematic wave (KW) theory any well-posed initial value traffic flow problem can be solved with the same methods either on the time–space (Euler) plane or the time vs vehicle number (Lagrange) plane. To achieve this symmetry the model parameters and the boundary data need to be expressed in a form appropriate for each plane. It turns out, however, that when boundary data that are bounded in one plane are transformed for the other, singular points with infinite density sometimes arise. Duality theory indicates that solutions to these problems must exist and be unique. Therefore, these solutions should be characterized.