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Higher-order Painlevé Equations in the Polynomial Class I. Bureau Symbol P2

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In this article, we construct all fourth- and fifth-order differential equations in the polynomial class having the Painlevé property and having the Bureau symbol P2. The fourth-order equations (including the Bureau barrier equation, y(iv)=3yy″−4(y′)2, which fails some Painlevé tests) are six in number and are denoted F-I,…,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 Painlevé transcendents. Other higher-order transcendents appearing here may be obtained by group-invariant reduction of the KdV5, Sawada–Kotera, and Kaup–Kupershmidt equations, the latter two being related. Four sections are devoted to solutions, first integrals, and assorted properties of the main equations. 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.
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Document Type: Original Article

Affiliations: The University of Sydney

Publication date: January 1, 2000

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