2014/07/06 by Hayato Chiba, Chiba, Hayato · 2 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.1407.1511
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 improve 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 (2m-th order first Painlevé equation).