A τ-function solution of the sixth painlevé transcendent

Abstract

We represent and analyze the general solution of the sixth Painlevé transcendent P6 in the Picard-HitchinOkamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ -functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the P6 equation and the uniformization of algebraic curves and present examples.

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

“…From the classification of Theorem B, such a λ(t) must be of the form λ r,s (t) with (r, s) ∈ Q N , N ≥ 3. Note that the problem concerning the distribution of poles of Painlevé VI solution has been addressed in [4,13,25]. Let φ(N) be the Euler function defined by…”

Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials

“…where δ 2 signifies an integer part of the number δ/2. These formulae are consequences of more general differential properties of Jacobi's functions briefly tabulated in [14]. One can see that the similar properties are inherent characteristics of the general Θ-functions if the g-dimensional jacobian is isomorphic to a product of elliptic curves.…”

The sixth Painlevé transcendent and uniformization of algebraic curves

“…See also comments as to work by Guzzetti [38] in Sect. 6 of work [15]. A direct check shows that the self-suggested generalization y = x s (x − 1) r does not exist for s, r / ∈ Q.…”

“…Derivation of the prime-forms (65) is just simplification. In order to present x(τ ) defined by (64) in form of such a ratio one needs to use the duplication formula θ 4 2 (z) − θ 4 1 (z) = ϑ 3 2 · θ 2 (2z), standard quadratic θ-identities [2,30,70], and second relation in (15).…”