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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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Original languageEnglish
Journal / PublicationJournal of Differential Equations
Online published7 Feb 2020
Publication statusOnline published - 7 Feb 2020

Abstract

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