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

TL;DR

This paper explores Darboux equations and their connection to Painlevé VI equations, introducing a new series expansion method and analyzing the correspondence and convergence of solutions.

Contribution

It establishes a full correspondence between special Darboux and Painlevé VI equations and introduces a novel infinite series expansion for solutions.

Findings

New series expansion in hypergeometric and elliptic functions

Full correspondence between special Darboux and Painlevé VI equations

Analysis of convergence of the series expansions

Abstract

In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlev\'e VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlev\'e VI equations. Instead of the 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 a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) %around each of the four regular singular points of the for the solutions of the Darboux equations and regard special solutions of the Darboux equations as those terminating series. The Darboux equations characterized in this manner have an almost (but not completely) full correspondence to the special types of the Painlev\'e VI equations. Finally, we discuss the convergence of these infinite series expansions.