Exact solutions for analog Hawking effect in dielectric media

In the framework of the analog Hawking radiation for dielectric media, we analyze a toy-model and also the 2D reduction of the Hopfield model for a specific monotone and realistic profile for the refractive index. We are able to provide exact solutions, which do not require any weak dispersion approximation. The theory of Fuchsian ordinary differential equations is the basic tool for recovering exact solutions, which are rigoroulsy identified, and involve the so-called generalized hypergeometric functions $_4F_3(\alpha_1,\alpha_2,\alpha_3,\alpha_4;\beta_1,\beta_2,\beta_3;z)$. A complete set of connection formulas are available, both for the subcritical case and for the transcritical one, and also the Stokes phenomenon occurring in the problem is fully discussed. From the physical point of view, we focus on the problem of thermality. Under suitable conditions, the Hawking temperature is deduced, and we show that it is in fully agreement with the expression deduced in other frameworks under various approximations.