  Versions of Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches
are applied to derive various partial differential equations (PDE) satisfied by
joint eigenvalue spacing probabilities of two coupled Gaussian Hermitian
matrices (coupled GUE). All the lowest (third) order PDE satisfied by the
probabilities for the largest eigenvalues of two coupled GUE are found, and the
results of both approaches are compared. The TW approach allows to derive all
PDE at once, while in the ASvM one starting with different bilinear identities
leads to different subsets of the full set of equations. An interesting result
is that the joint probability for the largest eigenvalues of coupled Gaussian
matrices satisfies a number of different PDE, and the previously known
Adler-van Moerbeke equation (AvM) [3] is only one of them. Some of the new
equations look like "coupled Painlev\'e IV" and have usual Painlev\'e IV
equation as one-matrix limit, i.e. when the spectral endpoint of one of the
matrices goes to infinity. This is in contrast to the AvM equation, which
becomes trivial in this limit. Moreover, the new PDE, which stem from the
matrix kernel approach of [23], do not contain derivatives w.r.t. the strength
of coupling, unlike the AvM equation. In other words, they contain fewer
independent variables and in this sense are simpler.
