Total integrals of global solutions to Painlevé II

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painlevé II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann–Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painlevé V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles that were obtained recently in (Baik et al 2008 Commun. Math. Phys. 280 463–97, Ehrhardt 2007 Commun. Math. Phys. 272 683–98). We also evaluate the total integrals of certain polynomials of the Painlevé functions and their derivatives. These polynomials are the densities of the first integrals of the modified Korteweg-de Vries equation. We discuss the relations of the formulae we have obtained to the classical trace formulae for the Dirac operator on the line