Connection problem of the first Painlevé transcendents with large initial data

In previous work, Bender and Komijani (2015 $\mathit{\text{J.}}\mathit{\text{Phys.}}\mathit{\text{A:}}\mathit{\text{Math.}}\mathit{\text{Theor.}}$ 48, 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a $\mathcal{P}\mathcal{T}$-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 $(y(0),y^{\mathrm{\prime }}(0))=(a,b)$ located on a sequence of curves ${\mathrm{\Gamma }}_n$, $n=1,2,\dots$, will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation $b^2\mathrm{/}4-a^3=f_n\sim A{n}^{6\mathrm{/}5}$ for the curves ${\mathrm{\Gamma }}_n$ as $n\to \mathrm{\infty }$ 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 ${\mathrm{\Gamma }}_n$ 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 $\mathit{\text{Arch.}}\mathit{\text{Rational}}\mathit{\text{Mech.}}\mathit{\text{Anal.}}$ {143}, 241–271) in the studies of the second Painlevé equation.