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

TL;DR

This paper analyzes the initial conditions leading to separatrix solutions of the first Painlevé equation, deriving asymptotic formulas for the curves separating different solution behaviors and confirming results with numerical data.

Contribution

It extends previous work by characterizing initial conditions on specific curves that produce separatrix solutions in a more general setting for the first Painlevé equation.

Findings

Derived asymptotic form of initial condition curves $\

Matched analytical formulas with numerical results for small $n$

Identified nonlinear eigenvalues $f_n$ governing solution types.

Abstract

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.