Self-similarity of the third type in ultra-relativistic blastwaves

I will introduce a new class of self-similar solutions that emerge in the problem of relativistic explosions. Consider an ultra-relativistic blastwave, propagating in a power-law density profile of the form rho ~ z^{-k}. Self-similar solutions of the first kind can be found for k < 7/4 using dimensional considerations. For steeper density gradients with k > 2, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate power-law indices 7/4 < k < 2, the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar dynamics are dictated by the non-self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non-self-similar flow is described by relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of a fast material that coincides with the interiors of the blastwave and terminates at the shock.