  The present paper studies a Gaussian Hermitian random matrix ensemble with
external source, given by a fixed diagonal matrix with two eigenvalues a and
-a. As a first result, the probability that the eigenvalues of the ensemble
belong to a set satisfies a fourth order PDE with quartic non-linearity; the
variables being the eigenvalue a and the boundary points of the set. This
equation enables one to find a PDE for the Pearcey distribution. The latter
describes the statistics of the eigenvalues near the closure of a gap; i.e.,
when the support of the equilibrium measure for large size random matrices has
a gap, which can be made to close. Precisely, the Gaussian Hermitian random
matrix ensemble with external source has this feature. In this work, we show
the Pearcey distribution satisfies a a fourth order PDE with cubic
non-linearity. The PDE for the finite problem is found by by showing that an
appropriate integrable deformation of the random matrix ensemble with external
source satisfies the three-component KP equation and Virasoro constraints.
