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Context. Calculations of highly energetic neutrino and TeV γ-ray emission from relativistic jets of Active Galactic Nuclei (AGN) are often based on a model that involves a collimated relativistic blast wave, in which the spectral evolution of energetic particles is determined by the interplay between the particle injection by sweep up of the interstellar medium, energy losses by radiation and diffusive escape. Aims. To date such models only have been solved numerically because of the highly non-linear nature of the time-dependent equations describing particle spectra and bulk energy loss. However it is difficult to see how parameters and groupings of parameters influence the resulting numerical solutions, except through intensive investigations of many numerical simulations. Therefore analytic solutions are particularly helpful. Methods. We provide exact mathematical solutions to a very broad class of AGN type models. By selecting different functional behaviors of parameter values, we cover a large compendium of possible situations. The comparison of the exact solutions with observational information can help to improve our understanding of the evolution of individual AGN. Exact solutions can also be used to provide controls on the appropropriateness and accuracy of numerical programs used to solve the equations. Results. We provide an analytical description of the evolution of proton spectra according to the pick-up model. We analyze the behavior of the particle spectra in the plasma frame. The solutions are determined by two competing processes: the deceleration of the jet plasmoid and particle cooling via radiation.

Channelled relativistic outflows in active galactic nuclei: analytic solutions for the evolution of particle spectra