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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we study the Darboux equations in both classical and systemform, which give the elliptic Painlev\'e VI equations by the isomonodromydeformation method. Then we establish the full correspondence between thespecial Darboux equations and the special Painlev\'e VI equations. Instead ofthe system form, we especially focus on the Darboux equation in a scalar form,which is the generalization of the classical Lam\'{e} equation. We introduce anew infinite series expansion (in terms of the compositions of hypergeometricfunctions and Jacobi elliptic functions) %around each of the four regularsingular points of the for the solutions of the Darboux equations and regardspecial solutions of the Darboux equations as those terminating series. TheDarboux equations characterized in this manner have an almost (but notcompletely) full correspondence to the special types of the Painlev\'e VIequations. Finally, we discuss the convergence of these infinite seriesexpansions.

Solutions of Darboux Equations, its Degeneration and Painlev\'e VI Equations