On the application of the Cole–Hopf transformation to hyperbolic equations based on second-sound models

We point out and examine two nonlinear, hyperbolic equations, both of which arise in kinematic-wave theory, that can be solved exactly using a conditional application of the Cole–Hopf transformation. Both of these equations are based on flux relations that were originally proposed as models of thermal wave phenomena, also known as second-sound. We then show how this method can be extended and used to obtain a particular type of exact solution to a class of nonlinear, hyperbolic PDEs.