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2 results

  • Solutions of the third Painlevé equation I

  • Hiroshi Umemura, Humihiko Watanabe
  • Journal:
    Nagoya Mathematical Journal / Volume 151 / September 1998
    Published online by Cambridge University Press:
    22 January 2016, pp. 1-24
    Print publication:
    September 1998
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  • We classify transcendental classical solutions of the third Painlevé equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painlevé equation.

  • Solutions of the second and fourth Painlevé equations, I

  • Hiroshi Umemura, Humihiko Watanabe
  • Journal:
    Nagoya Mathematical Journal / Volume 148 / December 1997
    Published online by Cambridge University Press:
    22 January 2016, pp. 151-198
    Print publication:
    December 1997
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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]).