  We propose a novel approach to the solution of nematic Liquid Crystal models
based on the derivation of a system of nonlinear wave equations for order
parameters such that the occurrence of uniaxial and biaxial phase transitions
can be interpreted as the propagation of a two-dimensional shock wave in the
space of thermodynamic parameters. We obtain the exact equations of state for
an integrable model of biaxial nematic liquid crystals and show that the
classical transition from isotropic to uniaxial phase in absence of external
fields is the result of a van der Waals type phase transition, where the jump
in the order parameters is a classical shock generated from a gradient
catastrophe at a non-zero isotropic field. The study of the equations of state
provides the first analytical description of the rich structure of nematics
phase diagrams in presence of external fields.
