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Global branching of solutions to ODEs and integrability

Rod Halburd

TL;DR

This work introduces and develops the algebro-Painlevé property, a generalization of the Painlevé property that allows finite branching when analytic continuation is performed on paths avoiding fixed singularities. It provides a complete classification of autonomous first-order ODEs with this property and derives necessary structural conditions for a broad second-order class with rational dependence on the dependent variable. Applying these tools to the Lane-Emden equation with index $n$, the authors prove that the algebro-Painlevé property holds precisely for $n\in\{0,1,5\}$, using a generalized alpha-test and a careful analysis of movable singularities. The results offer a robust framework for identifying integrable cases, linking to Riccati reductions, elliptic-function representations, and known integrable reductions such as the Takasaki system, and provide general structural theorems for a wide second-order family $w''=L(z,w)(w')^2+M(z,w)w'+N(z,w)$.

Abstract

We consider a natural generalisation of the Painlevé property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with this property and find necessary conditions for a large family of second-order equations. We consider ODEs such that, given any simply connected domain $Ω$ not containing fixed singularities of the equation, the Riemann surface of any solution obtained by analytic continuation along curves in $Ω$ has a finite number of sheets over $Ω$.

Global branching of solutions to ODEs and integrability

TL;DR

This work introduces and develops the algebro-Painlevé property, a generalization of the Painlevé property that allows finite branching when analytic continuation is performed on paths avoiding fixed singularities. It provides a complete classification of autonomous first-order ODEs with this property and derives necessary structural conditions for a broad second-order class with rational dependence on the dependent variable. Applying these tools to the Lane-Emden equation with index , the authors prove that the algebro-Painlevé property holds precisely for , using a generalized alpha-test and a careful analysis of movable singularities. The results offer a robust framework for identifying integrable cases, linking to Riccati reductions, elliptic-function representations, and known integrable reductions such as the Takasaki system, and provide general structural theorems for a wide second-order family .

Abstract

We consider a natural generalisation of the Painlevé property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with this property and find necessary conditions for a large family of second-order equations. We consider ODEs such that, given any simply connected domain not containing fixed singularities of the equation, the Riemann surface of any solution obtained by analytic continuation along curves in has a finite number of sheets over .

Paper Structure

This paper contains 5 sections, 2 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. First-order equations
  3. The Lane-Emden Equation
  4. A general family of equations
  5. Conclusion

Key Result

Lemma 2.1

Let $f$ be such that $f'(z)=(z-a)^{-r}g(z)$, where $r\ge 1$ is a rational number, $g$ is holomorphic on the disc $D=\{z\in\mathbb{C}:0<|z-a|<\epsilon\}$ of radius $\epsilon>0$ centred at $a$, and $g(a)\ne 0$. If $r>1$ then given any $\xi\in\mathbb{C}\setminus\{0\}$ there is a curve $\gamma:[0,1)\to

Theorems & Definitions (2)

  • Lemma 2.1
  • Lemma 2.3