Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field

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5 Citations (Scopus)

Abstract

Formal series solutions and the Kovalevskaya exponents of a quasi-homogeneous polynomial system of differential equations are studied by means of a weighted projective space and dynamical systems theory. A necessary and sufficient condition for the series solution to be a convergent Laurent series is given, which improves the well-known Painlevé test. In particular, if a given system has the Painlevé property, an algorithm to construct Okamoto's space of initial conditions is given. The space of initial conditions is obtained by weighted blow-ups of the weighted projective space, where the weights for the blow-ups are determined by the Kovalevskaya exponents. The results are applied to the first Painlevé hierarchy (2. m-th order first Painlevé equation).

Original languageEnglish
Pages (from-to)7681-7716
Number of pages36
JournalJournal of Differential Equations
Volume259
Issue number12
DOIs
Publication statusPublished - 2015 Dec 15
Externally publishedYes

Keywords

  • Kovalevskaya exponent
  • Quasi-homogeneous vector field
  • The first Painlevé hierarchy
  • Weighted projective space

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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