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

Abstract

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 $$s_1=-e^{-2\alpha \pi i},s_2=\omega ,s_3=-e^{2\alpha \pi i}$$ with $\omega$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $\omega =0$. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for $\alpha >-1/2$ and $\omega \ge 0$, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters $\alpha$ and $\omega$ are chosen to be special values. Some applications of our results in random matrix theory are also discussed.

41 pages, 6 figures, references updated