Abstract
We consider a family of tronquée solutions of the Painlevé II equationq″(s)=2q(s)3+sq(s)−(2α+12),α>−12, which is characterized by the Stokes multiplierss1=−e−2απi,s2=ω,s3=−e2απi 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.
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