# Higher-order Painleve equations in the polynomial class I. Bureau symbol P2

## Author

**Christopher M. Cosgrove**

## Status

Research Report 98-22

Date: 24 June 1998, revised 30 July 1998

## Abstract

In this paper, we construct all fourth- and fifth-order
differential equations in the polynomial class having
the Painleve property and having the Bureau symbol P2.
The fourth-order equations (including the Bureau barrier
equation, y'''' = 3yy'' - 4(y')^2, which fails some
Painleve tests) are six in number and are denoted F-I,
F-II, ..., F-VI; the fifth-order equations are four in
number and are denoted Fif-I, ..., Fif-IV. The 12 remaining
equations of the fourth order in the polynomial class
(where the Bureau symbol is P1) are listed in the Appendix,
their proof of uniqueness being postponed to a sequel
(paper II).
Earlier work on this problem by Bureau, Exton,
and Martynov is incomplete, Martynov having found 13 of
the 17 distinct reduced equations. Equations F-VI and
Fif-IV are new equations defining new higher-order
Painleve transcendents. Other higher-order transcendents
appearing here may be obtained by group-invariant reduction
of the KdV5, Sawada-Kotera, and Kaup-Kuperschmidt equations,
the latter two being related. Four sections are devoted to
solutions, first integrals, properties, etc., and a section
is devoted to the Bureau barrier equation, whose integrability
status is unknown, and a new hierarchy built on this equation.
Several of the equations are solved in terms of hyperelliptic
functions of genus 2 by means of Jacobi's postmultiplier
theory. Except for a classic solution of Drach, we believe
that all of these hyperelliptic solutions are new. In an
accompanying paper, the hyperelliptic solutions of F-V and
F-VI are applied to the unsolved third-order Chazy classes
IX and X.

## Key phrases

Painleve analysis. Painleve transcendents. ordinary differential
equations. hyperelliptic functions. nonlinear equations.

## AMS Subject Classification (1991)

Primary: 34A34

Secondary: 34A05, 34A20, 33E30

## Content

The paper is available in the following forms:
- TeX dvi format:
- 1998-22.dvi.gz (106kB) or
1998-22.dvi (293kB)

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- 1998-22.ps.gz (169kB) or
1998-22.ps (533kB)

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