Lie families: theory and applications
arXiv:1003.3529 · doi:10.1088/1751-8113/43/30/305201
Abstract
We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.
23 pages, revised version to appear in J. Phys. A: Math. Theor
References in corpus (11)
- Superposition rules, Lie theorem and partial differential equations
- Nonlinear Shear-free Radiative Collapse
- New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems
- A nonlinear superposition rule for solutions of the Milne--Pinney equation
- Recent Applications of the Theory of Lie Systems in Ermakov Systems
- Applications of Lie systems in dissipative Milne--Pinney equations
- Quasi-Lie schemes: theory and applications
- Quasi-Lie schemes and Emden--Fowler equations
- U(1)-invariant membranes: the geometric formulation, Abel and pendulum differential equations
- A geometric approach to integrability conditions for Riccati equations
- Symmetry, Equivalence and Integrable Classes of Abel Equations
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- 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)
- Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation
- Superposition rules and second-order Riccati equations
- Stratified Lie systems: Theory and applications