Painleve Transcendents and PT-Symmetric Hamiltonians

TL;DR

This paper investigates the asymptotic behavior of separatrix solutions for Painlevé I and II transcendents, linking their properties to PT-symmetric Hamiltonians and deriving key constants analytically and numerically.

Contribution

It provides a novel analytical derivation of the asymptotic constants for Painlevé transcendents using PT-symmetric quantum Hamiltonian eigenvalue problems.

Findings

Asymptotic formulas for initial slopes and values of Painlevé I and II solutions.

Numerical determination of constants B_I, C_I, B_II, C_II.

Analytical expressions for these constants via PT-symmetric Hamiltonians.

Abstract

Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y'(0)= 0$, there is a discrete set of initial values $y(0)=c_n$ that give rise to separatrix solutions. For Painlev\'e I the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\rm I}n^{3/5}$ and that of $c_n$ is $c_n\sim C_{\rm I}n^{2/ 5}$, and for Painlev\'e II the large-$n$ asymptotic behavior of $b_n$ is $b_n \sim B_{\rm II}n^{2/3}$ and that of $c_n$ is $c_n\sim C_{\rm II}n^{1/3}$. The constants $B_{\rm I}$, $C_{\rm I}$, $B_{\rm II}$, and $C_{\rm II}$ are first determined numerically. Then, they are found analytically and in closed form by reducing the nonlinear equations to the linear eigenvalue problems associated with the cubic and quartic PT-symmetric Hamiltonians $H=\frac{1}{2}p^2+2ix^3$ and $H=\frac{1}{2}p^2-\frac{1}{2}x^4$.