Schwarzian derivative in higher-order Riccati equations

TL;DR

This paper explores the role of Schwarzian derivatives in higher-order Riccati equations, revealing that these derivatives naturally extend to and embed within the Riccati hierarchy, thus generalizing their connection beyond first order.

Contribution

It introduces higher-order analogues of the Schwarzian derivative and demonstrates their embedding in the Riccati hierarchy, extending the classical connection to higher-order equations.

Findings

Higher-order Schwarzian derivatives exist and are meaningful.

Riccati equations of order two and three are embedded in these derivatives.

The classical connection between Schwarzian and Riccati extends to higher orders.

Abstract

The Sturm-Liouville equation represents the linearized form of the first-order Riccati equation. This provides an evidence for the connection between Schwarzian derivative and this first-order nonlinear differential equation. Similar connection is not obvious for higher-order equations in the Riccati chain because the corresponding linear equations are of order greater than two. With special attention to the second- and third-order Riccati equations we demonstrate that Schwarzian derivative has a natural space in higher Riccati equations. There exist higher-order analogues of the Schwarztan derivative. We demonstrate that equations in the Riccati hierarchy are embedded in these higher-order derivatives.