On integrals of the tronqu\'{e}e solutions and the associated Hamiltonians for the Painlev\'{e} II equation

We consider a family of tronqu\'{e}e solutions of the Painelv\'{e} II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2\alpha+\frac12), \qquad \alpha > -\frac12, \end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2\alpha \pi i },\quad s_2=\omega, \quad s_3=-e^{2 \alpha \pi i}$$ 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 tronqu\'{e}e solutions and the associated Hamiltonians over the real axis for $\alpha > -1/2$ and $\omega \geq 0$, 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.