Exact meromorphic solutions of Schwarzian differential equations

TL;DR

This paper explicitly constructs all transcendental meromorphic solutions of five types of autonomous Schwarzian differential equations, revealing that four types' solutions are elliptic functions and characterizing solutions with special properties.

Contribution

It provides explicit constructions of solutions for five canonical types of Schwarzian equations and characterizes solutions with injectivity or Picard exceptional values.

Findings

Solutions of four types are elliptic functions

Explicit solutions are constructed for five types

Solutions with injectivity or Picard exceptional values are characterized

Abstract

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type.