Mixed superposition rules for Lie systems and compatible geometric structures
Rutwig Campoamor-Stursberg, Oscar Carballal, Francisco J. Herranz, Javier de Lucas
TL;DR
The paper develops a geometric-coalgebra framework for mixed superposition rules in Lie systems, enabling the general solution to be reconstructed from particular solutions of auxiliary systems via maps $\Phi$ on product manifolds and momentum maps. It extends the coalgebra method to Dirac–Lie and Lie–Hamilton (LH) systems, addresses topological issues by using Fréchet spaces and well-behaved topological Poisson coalgebras, and provides criteria for imprimitive and semidirect-sum VG algebras to admit natural mixed rules. The authors derive explicit mixed rules for Schrödinger LH systems (e.g. $S(1)$), Riccati problems, and time-dependent Calogero–Moser systems, and illustrate applications to time-dependent thermodynamics and PDE Lie systems such as the Tzitzéica equation. Overall, the work broadens the solvability toolkit by unifying Lie theory with Dirac geometry and Poisson coalgebras, enabling reduced-system solutions to yield general solutions across finite- and infinite-dimensional settings.
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 Schrödinger 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 Tzitzéica equation and a related system is given.