# Nonlinear eigenvalue problems for generalized Painlev\'e equations

## TL;DR

This paper extends the concept of eigenvalue problems to nonlinear differential equations, specifically generalized Painlevé equations, revealing discrete spectra and novel phenomena like hyperfine splitting through analytical and numerical analysis.

## Contribution

It introduces a broad class of nonlinear differential equations generalizing Painlevé equations that possess discrete eigenvalue spectra and analyzes their large-eigenvalue behavior.

## Key findings

- Discrete eigenvalue spectra for generalized Painlevé equations
- Identification of hyperfine splitting of eigenvalues
- Analytical and numerical characterization of eigenvalue behavior

## Abstract

Eigenvalue problems for linear differential equations, such as time-independent Schr\"odinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation $y'(x)=\cos[\pi xy(x)]$ and the first, second, and fourth Painlev\'e transcendents. It is shown here that the differential equations for the first and second Painlev\'e transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively.