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

We consider a family of tronquée solutions of the Painelvé II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2α+\frac12), \qquad α> -\frac12, \end{equation*} which is characterized by the Stokes multipliers $s_1=-e^{-2απi },\quad s_2=ω, \quad s_3=-e^{2 απi}$ 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 $ω\geq 0$, 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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