  Airy and Pearcey-like kernels and generalizations arising in random matrix
theory are expressed as double integrals of ratios of exponentials, possibly
multiplied with a rational function. In this work it is shown that such kernels
are intimately related to wave functions for polynomial (Gel'fand-Dickey
reductions) or rational reductions of the KP-hierarchy; their Fredholm
determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a
systematic way, non-linear PDEs for the Fredholm determinant of such kernels.
Examples include Fredholm determinants giving the gap probability of some
infinite-dimensional diffusions, like the Airy process, with or without
outliers, and the Pearcey process, with or without inliers.
