Hamiltonian system for the elliptic form of Painlevé VI equation

Abstract

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP 1 . In this paper, we extend this isomonodromy theory on CP 1 to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in CP 1 . This is one of the advantages of the elliptic form. u zz − 1 2 u 2 z z = (u zz ) z − u z u zz = − 1 4 e u z + 1 4 e u u z = 0.

Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

“…This fact might be indirectly proved by the transformation (1.2). In [5], this fact and the associated Hamiltonian system has been directly derived. Naturally, this isomonodromic feature proposes the following question:…”

New existence results for the mean field equation on compact surfaces via degree theory

“…More recently, equation (1.1) has been studied in the context of hyperelliptic curves and of the Painlevé equations, see [9] and [11], respectively. Equation (1.1) plays also an important role in mathematical physics.…”

On reducible monodromy representations of some generalized Lamé equation

“…By (2.4) there are two cases. Case 1. a j ∈ E τ [2] for j = 1, 2, then a 2 = −a 1 . Then (r, s) = (0, 0), i.e.…”

“…Indeed, if (A(τ), B(τ), p(τ)) depends on τ suitably such that GLE (1.2) preserves the monodromy as τ deforms, then p(τ) satisfies the elliptic form of Panlevé VI equation. See [2] or Section 3. Note that by letting x = ℘(z), GLE (1.2) can be projected to a new equation on CP 1 , which is a second order Fuchsian equation with five singular points {e 1 , e 2 , e 3 , ℘(p), ∞} with ℘(p) being apparent.…”