  We present a method to study phase transitions in the large N limit of matrix
models using matched solutions of Whitham hierarchies. The endpoints of the
eigenvalue spectrum as functions of the temperature are characterized both as
solutions of hodograph equations and as solutions of a system of ordinary
differential equations. In particular we show that the free energy of the
matrix model is the quasiclassical tau-function of the associated hierarchy,
and that critical processes in which the number of cuts changes in one unit are
third-order phase transitions described by C1 matched solutions of Whitham
hierarchies. The method is illustrated with the Bleher-Eynard model for the
merging of two cuts. We show that this model involves also a birth of a cut.
