Schwarzian derivative, Painlev\'e XXV-Ermakov equation and B\"acklund transformations

TL;DR

This paper explores how the Schwarzian derivative underpins solutions and invariances of the Painlevé XXV-Ermakov equation and related linear equations, introducing Bäcklund transformations that connect linear and nonlinear forms.

Contribution

It demonstrates the central role of the Schwarzian derivative in constructing solutions and invariances for complex differential equations, and introduces new Bäcklund transformations linking linear and nonlinear equations.

Findings

Solutions based on Schwarzian derivatives are key to understanding Painlevé XXV-Ermakov equations.

Two families of Bäcklund transformations are derived, connecting linear and nonlinear equations.

Examples illustrate applications of the theoretical results.

Abstract

The role of Schwarzian derivative in the study of nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlev\'e XXV-Ermakov equation, Ermakov equation and third order linear equation in a normal form are shown to be based on solutions of the Schwarzian equation. Starting from the Riccati equation and the second order element of the Riccati chian as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of B\"acklund transformations which link the linear and nonlinear equations under investigation are obtained. Some examples with relevant applications are given and discussed.