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


Christopher M. Cosgrove


Research Report 98-22
Date: 24 June 1998, revised 30 July 1998


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


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