Mixed superposition rules for Lie systems and compatible geometric structures

TL;DR

This paper develops new methods for solving Lie systems using mixed superposition rules linked to geometric structures, with applications to physical systems and PDEs, extending existing coalgebra techniques.

Contribution

It introduces a novel mixed coalgebra method for Lie systems on Dirac structures and extends superposition rules to PDEs, broadening the applicability of Lie system solutions.

Findings

Derived practical methods for mixed superposition rules

Extended coalgebra method to Hamiltonian Lie systems

Applied results to physical and PDE systems

Abstract

Mixed superposition rules are, in short, a method to describe the general solutions of a time-dependent system of first-order differential equations, a so-called Lie system, in terms of particular solutions of other ones. This article is concerned with the theory of mixed superposition rules and their connections with geometric structures. We provide methods to obtain mixed superposition rules for systems admitting an imprimitive finite-dimensional Lie algebra of vector fields or given by a semidirect sum. In particular, we develop a novel mixed coalgebra method for Lie systems that are Hamiltonian relative to a Dirac structure, which is quite general, although we restrict to symplectic and contact manifolds in applications. This provides us with practical methods to derive mixed superposition rules and extends the coalgebra method to a new field of application while solving minor technical issues of the known formalism. Throughout the paper, we apply our results to physical systems including Schr\"odinger Lie systems, Riccati systems, time-dependent Calogero-Moser systems with external forces, time-dependent harmonic oscillators, and time-dependent thermodynamical systems, where general solutions can be obtained from reduced system solutions. Our results are finally extended to Lie systems of partial differential equations and a new source of such PDE Lie systems, related to the determination of approximate solutions of PDEs, is provided. An example based on the Tzitz\'eica equation and a related system is given.