  We show that the derivative of the logarithm of the average characteristic
polynomial of a diffusing Wishart matrix obeys an exact partial differential
equation valid for an arbitrary value of N, the size of the matrix. In the
large N limit, this equation generalizes the simple Burgers equation that has
been obtained earlier for Hermitian or unitary matrices. The solution through
the method of characteristics presents singularities that we relate to the
precursors of shock formation in fluid dynamical equations. The 1/N corrections
may be viewed as viscous corrections, with the role of the viscosity being
played by the inverse of the doubled dimension of the matrix. These corrections
are studied through a scaling analysis in the vicinity of the shocks, and one
recovers in a simple way the universal Bessel oscillations (so-called hard edge
singularities) familiar in random matrix theory.
