The second member of the Painlevé I hierarchy, \(P_I^2\), is a fourth order nonlinear ODE in \(x\) depending on the parameter \(t\). This equation governs isomonodromic deformations in \(x\) and \(t\) of a linear \(2\times2\) matrix linear ODE in \(\zeta\) with a 3rd degree polynomial coefficients nilpotent in the leading in \(\zeta\) order. Thus the space of solutions to \(P_I^2\) can be parameterized by four (of totally seven) Stokes multipliers, \(s_k\).
The author considers the particular unique solution \(y(x,t)\) to \(P_I^2\) characterized by the values \(s_1=s_2=s_5=s_6=0\). This solution can be understood as an analog of the famous tritronqueé solution to the first Painlevé equation \(P_I\). Using the steepest descent analysis of the corresponding Riemann-Hilbert problem, he obtains the asymptotic behavior of \(y(x,t)\) as \(x,t\to\pm\infty\) while \(s=xt^{-3/2}\in{\mathbb R}\) remains bounded. The author proves that \(y(x,t)\) has an algebraic asymptotics for \(s\in{\mathbb R}\backslash[-2\sqrt3, \tfrac{2\sqrt{15}}{27}]\). In the interior of the excluded interval, \(y(x,t)\) has an elliptic asymptotic behavior. Also, without much details, the author provides the critical asymptotics of \(y(x,t)\) corresponding to the boundaries of the above interval: in terms of the Hastings-McLeod solution to the second Painlevé equation \(P_{I\!I}\) for \(s=-2\sqrt3\), and in terms of hyperbolic functions for \(s=\tfrac{2\sqrt{15}}{27}\).