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Solutions of the second and fourth Painlevé equations, I

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura  and
Humihiko Watanabe
Hiroshi Umemura
Affiliation:
Graduate School of Polymathematics, Nagoya University, Nagoya 464-01, Japan, [email protected]
Humihiko Watanabe
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 810, Japan
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Abstract

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A rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 148 , December 1997 , pp. 151 - 198
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

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