On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

We consider a family of tronquée solutions of the Painlevé II equation q′′ (s) = 2q (s)³ + sq (s)−(2α+½), α >−½, which is characterized by the Stokes multipliers s₁=−e^−2απi, s₂=ω, s₃ = −e^2απi with ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω = 0. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for α>−1/2 and ω≥0, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed.