TY - JOUR

T1 - Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field

AU - Chiba, Hayato

N1 - Funding Information:
The author would like to thank Professor Yasuhiko Yamada for useful comments. This work was supported by Grant-in-Aid for Young Scientists (B), No. 25800081 from MEXT Japan.
Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/12/15

Y1 - 2015/12/15

N2 - 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).

AB - 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).

KW - Kovalevskaya exponent

KW - Quasi-homogeneous vector field

KW - The first Painlevé hierarchy

KW - Weighted projective space

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U2 - 10.1016/j.jde.2015.08.035

DO - 10.1016/j.jde.2015.08.035

M3 - Article

AN - SCOPUS:84943357009

SN - 0022-0396

VL - 259

SP - 7681

EP - 7716

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 12

ER -