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

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

4 Scopus Citations
View graph of relations


Related Research Unit(s)


Original languageEnglish
Pages (from-to)2430-2476
Journal / PublicationJournal of Differential Equations
Issue number3
Online published7 Feb 2020
Publication statusPublished - 15 Jul 2020


We consider a family of tronquée solutions of the Painlevé II equation q′′ (s) = 2(s)+ s(s)−(2α+½), α >−½, which is characterized by the Stokes multipliers s1=−e−2απis2=ω, s= −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.

Research Area(s)

  • Asymptotic expansion, Painlevé equation, Random matrix theory, Riemann-Hilbert method

Citation Format(s)