PVi

Painlevé VI, abbreviated pVI, is the sixth and most general equation in the classical Painlevé family of nonlinear second-order ordinary differential equations. Like the other Painlevé equations, pVI is notable for possessing the Painlevé property, meaning its solutions have no movable branch points or essential singularities; their only movable singularities are poles. pVI depends on four parameters and plays a central role in the theory of isomonodromic deformations and integrable systems.

The standard form of Painlevé VI involves a dependent variable y(t) with independent variable t, and four constants α, β, γ, δ. It is written as

y'' = 1/2 [1/y + 1/(y-1) + 1/(y-t)] (y')^2

- [1/t + 1/(t-1) + 1/(y-t)] y'

+ [ y(y-1)(y-t) / (t^2 (t-1)^2) ] [ α + β/y + γ/(y-1) + δ/(y-t) ],

where primes denote differentiation with respect to t.

Conceptually, pVI arises as the isomonodromic deformation of a 2×2 Fuchsian system with four regular singularities at 0, 1, t, and ∞. As t varies, the monodromy data of the associated linear system are preserved, and the corresponding nonlinear equation governs the motion of a movable singularity y(t). The theory introduces the tau function, Bäcklund transformations, and rich symmetry structures that generate families of solutions.

pVI has diverse specializations and limits: for certain parameter choices it reduces to algebraic or classical special-function solutions; in degenerate limits it connects to the other Painlevé equations (II–V). It was studied in depth by Painlevé and Gambier in the early 20th century and remains central in mathematical physics and differential geometry.