Lie systems: theory, generalisations, and applications

Abstract

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.

161 pages, 2 figures

Cited by in corpus (36)

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• Symmetry of stochastic non-variational differential equations
• A class of exact solutions of the Liénard type ordinary non-linear differential equation
• From constants of motion to superposition rules for Lie-Hamilton systems
• Dirac--Lie systems and Schwarzian equations
• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Lie-Hamilton systems on curved spaces: A geometrical approach
• Wei-Norman equations for a unitary evolution
• Lie symmetries for Lie systems: applications to systems of ODEs and PDEs
• A Lie systems approach to the Riccati hierarchy and partial differential equations
• Mixed superposition rules and the Riccati hierarchy
• On Lie systems and Kummer-Schwarz equations
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Contact Lie systems
• Multisymplectic structures and invariant tensors for Lie systems
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Geometry of Riccati equations over normed division algebras
• Jacobi-Lie systems: Fundamentals and low-dimensional classification
• Quasi-Lie schemes for PDEs
• Quasi-Lie families, schemes, invariants and their applications to Abel equations
• A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)
• Quantum quasi-Lie systems: properties and applications
• Quasi-rectifiable Lie algebras for partial differential equations
• A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting
• A new application of k-symplectic Lie systems
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Geometric numerical methods for Lie systems and their application in optimal control
• Stratified Lie systems: Theory and applications
• Hamiltonian stochastic Lie systems and applications
• Applications of standard and Hamiltonian stochastic Lie systems
• Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras
• An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold
• Reduction and reconstruction of multisymplectic Lie systems