The Elliptic Representation of the General Painlevé 6 Equation

Singular normal form for the Painlevé equation P1

We show that there exists a rational change of coordinates of Painlevé's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singula…

Painlevé第六方程的最小代数解

Minimal algebraic solutions of the sixth equation of Painlevé

本文针对Painlevé第六方程的48个例外代数解，构造了次数推测为最小的代数曲线并给出最优参数表示，发现除15个解外其次数等于分支数。

固定参数下 Painlevé III 方程特殊函数解的渐近性质

Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters

本文利用 Bessel 函数的 Toeplitz 行列式表示与多重围道积分，计算了复平面上 Painlevé-III 方程特殊函数解的小 $x$ 和大 $x$ 渐近性质，并指出该表示可用于数值计算。

奇异椭圆曲线的邻域与无穷远处的黎曼-希尔伯特对应

Neighborhoods of singular elliptic curves and the Riemann-Hilbert correspondence at infinity

本文研究奇异椭圆曲线邻域的解析分类，通过双重覆盖与光滑情形不变量关联，并推测其与Painlevé VI的无穷远黎曼-希尔伯特对应有关。