# Higher-order Painleve equations in the polynomial class II. Bureau symbol P1

## Author

**Christopher M. Cosgrove**

## Status

Research Report 2000-6

Date: 16 November 1999, revised 21 February 2000

## Abstract

In this paper, we complete the Painleve classification of fourth-order
differential equations in the polynomial class that was begun in paper I,
where the subcase having Bureau symbol P2 was treated. This paper treats the
more difficult subcase having Bureau symbol P1. Some of the calculations
involve the use of computer searches to find all cases of integer resonances.
Other cases are better handled with the Conte-Fordy-Pickering test.
The final list consists of 18 equations denoted F-I, F-II, ..., F-XVIII, built
upon 17 distinct reduced equations, one of which (F-II) is not a Painleve-type
equation. The last 12 beginning with F-VII have Bureau symbol P1, and the
main task of this paper is to prove that there are no others. Equations F-V,
F-VI, F-XVII, and F-XVIII define higher-order Painleve transcendents. Of
these, F-VI is new while the other three are group-invariant reductions of the
KdV5, the Modified KdV5, and the Modified Sawada-Kotera equations,
respectively. Seven of the 18 equations involve hyperelliptic functions of
genus 2. We also add to the list another interesting non-Painleve equation,
belonging to a hierarchy of Clarkson and Olver, and give it the number F-XIX.

Partial results on the fourth-order classification problem have been obtained
previously by Bureau, Exton, and Martynov, the latter author obtaining 13 of
the 17 distinct reduced equations with the Painleve property. Complete
solutions are given except in the cases that define the aforementioned
higher-order transcendents.

## Key phrases

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

## AMS Subject Classification (1991)

Primary: 34A34

Secondary: 34A05, 34A20, 33E20

## Content

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