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

In previous work, Bender and Komijani (2015 \textit{J. Phys. A: Math. Theor.} 48, 475202) studied the first Painlev\'e (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a $\mathcal{PT}$-symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions $(y(0),y'(0))=(a,b)$ located on a sequence of curves $\Gamma_n$, $n=1,2,\dots$, will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation $b^2/4 - a^3=f_n \sim A n^{6/5}$ for the curves $\Gamma_{n}$ as $n\to\infty$ is derived, where $A$ is a positive constant. The discrete set $\{f_n\}$ could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of $\Gamma_n$ matches the numerical results remarkably well, even for small $n$. The main tool is the method of uniform asymptotics introduced by Bassom et al. (1998 \textit{Arch. Rational Mech. Anal.} {143}, 241--271) in the studies of the second Painlev\'e equation.