On Lie systems and Kummer-Schwarz equations

Abstract

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer--Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne--Pinney and other related equations. We demonstrate that all the above-mentioned equations associated with exactly the same Lie system on SL(2,R) can be integrated simultaneously. This retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.

29 pages. A relevant error and several typos corrected

References in corpus (6)

• Superposition rules, Lie theorem and partial differential equations
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• Mixed superposition rules and the Riccati hierarchy
• Explicit solutions of the ${\mathfrak{a}}_1$-type Lie-Scheffers system and a general Riccati equation
• Dynamical Studies of Equations from the Gambier Family
• Local and Global Aspects of Lie's Superposition Theorem

Cited by in corpus (2)

• k-symplectic Lie systems: theory and applications
• Poisson-Hopf algebra deformations of Lie-Hamilton systems