Dirac--Lie systems and Schwarzian equations

TL;DR

This paper introduces Dirac--Lie systems on Dirac manifolds, providing new methods to analyze their properties and applying these to Schwarzian equations and other physics-related differential equations.

Contribution

It develops the theory of Dirac--Lie systems, extending Lie system analysis to Dirac manifolds, and demonstrates applications to Schwarzian equations and physics.

Findings

Enhanced analysis of constants of motion and superposition rules.

Application of Dirac geometry to differential equations in physics.

Effective methods for studying traveling wave solutions.

Abstract

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the Dirac's description of constrained systems, we introduce and analyse a particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems, which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this `Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze traveling wave solutions of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics.