English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In literature, it is known that any solution of Painlev\'{e} VI equationgoverns the isomonodromic deformation of a second order linear Fuchsian ODE on$\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on$\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying theisomonodromic deformation of the generalized Lam\'{e} equation. Among otherthings, we prove that the isomonodromic equation is a new Hamiltonian system,which is equivalent to the elliptic form of Painlev\'{e} VI equation forgeneric parameters. For Painlev\'{e} VI equation with some special parameters,the isomonodromy theory of the generalized Lam\'{e} equation greatly simplifiesthe computation of the monodromy group in $\mathbb{CP}^{1}$. This is one of theadvantages of the elliptic form.

Hamiltonian system for the elliptic form of Painlev\'{e} VI equation