In physics, the ferromagnetic Curie-Weiss spin model leads to a transcendental self-consistency equation for the magnetization $m$ of the form $$ m = \tanh(J m +h)\ , $$ with $J>0$ and $h\in\mathbb{R}$. This equation can have one or three solutions, depending on the value given to the parameters $(J,h)$. The regions in the $(J,h)$ plane where one or three solutions are expected are shown (schematically) in the picture below: for $J>1$ and $|h|$ small (shaded area), we expect three solutions, and one solution everywhere else. I am interested in a more precise characterization of the boundary curves: is it possible to describe analytically (even in the form of an implicit equation) the two boundary curves - the loci of the transition between one and three solutions of the self-consistency equation in the $(J,h)$ plane? If not, I would also be happy with an elegant solution, say, in Mathematica for this problem, as I have not been able to find a satisfactory answer so far. Many thanks in advance for any help.
