Superposition rules for higher-order systems and their applications

TL;DR

This paper extends the concept of superposition rules to higher-order differential systems, proving their existence and deriving new rules for specific equations, with applications in physics and mathematics.

Contribution

It introduces a generalized framework for superposition rules in higher-order systems and presents novel rules for Kummer--Schwarz equations.

Findings

Existence of superposition rules for higher-order systems

New superposition rules for second- and third-order Kummer--Schwarz equations

Illustrative examples from physics and mathematics literature

Abstract

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations and analyse their properties. Several results concerning the existence of various types of superposition rules for higher-order systems are proved and illustrated with examples extracted from the physics and mathematics literature. In particular, two new superposition rules for second- and third-order Kummer--Schwarz equations are derived.