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In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painleve I equation. We will show that this system passes the Painleve $alpha$-test for all accessible singular points $P_i (i=1,2,3)$. We note its holomorphy condition of the first Painleve system.
We construct the initial-value space of a $q$-discrete first Painleve equation explicitly and describe the behaviours of its solutions $w(n)$ in this space as $ntoinfty$, with particular attention paid to neighbourhoods of exceptional lines and irred
A Lax formalism for the elliptic Painleve equation is presented. The construction is based on the geometry of the curves on ${mathbb P}^1times{mathbb P}^1$ and described in terms of the point configurations.
Bilinear structure for the discrete Painleve I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painleve I equation is naturally
We consider the generalized Painleve--Ince equation, begin{equation*} ddot{x}+alpha xdot{x}+beta x^{3}=0 end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are relate
We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequaliti