Connection problem of the first Painlev\'{e} transcendents with large initial data

In previous work, Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) showed that the separatrix solutions to the first Painlev\'e (PI) equations are determined by a $\mathcal{PT}$-symmetric Hamiltonian, and a sequence of initial conditions give rise to separatrix solutions. In the present work, we consider the initial value problem of the PI equation in a more general setting and show that the initial conditions $y(0)$ and $y'(0)$ located on a sequence of curves $\Gamma_n$, $n=1,2,\dots$, will give rise to separatrix solutions, and we find the limiting form equation of the curves $\Gamma_{n}$ as $n\to\infty$. These curves separate the singular solutions and oscillating solutions of PI. Our analytical asymptotic formula of $\Gamma_n$ match the numerical results remarkably well, even for small $n$.