  We show that Hermitian matrix models support the occurrence of a new type of
phase transition characterised by dispersive regularisation of the order
parameter near the critical point. Using the identification of the partition
function with a solution of a reduction of the Toda hierarchy, known as
Volterra system, we argue that the singularity is resolved by the onset of a
multi-dimensional dispersive shock of the order parameter in the space of
coupling constants. This analysis explains the origin and mechanism leading to
the emergence of chaotic behaviours observed in M^6 matrix models and extends
its validity to even nonlinearity of arbitrary order.
