Predicted travelling pulse solutions in the diffusion approximation (Eq. (21), thin curve) and in the telegraph equation model (Eq. (26), thick curve) for t = 20. The parameters are ξ =  − 1 and τ = 1. The solution to the telegraph equation vanishes for and, in contrast to that of the diffusion equation, has an asymmetrical shape that leads to shorter rise times for the intensity of solar energetic particles.