Mixed superposition rules and the Riccati hierarchy
arXiv:1203.0123 · doi:10.1016/j.jde.2012.08.020
Abstract
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.
20 pages
References in corpus (16)
- Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space
- Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
- Superposition rules, Lie theorem and partial differential equations
- Lie systems: theory, generalisations, and applications
- 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
- Lie--Hamilton systems: theory and applications
- Superposition rules for higher-order systems and their applications
- Periodic first integrals for Hamiltonian systems of Lie type
- Quasi-Lie schemes: theory and applications
- Lie families: theory and applications
- A dynamic correspondence between Bose-Einstein condensates and Friedmann-Lemaître-Robertson-Walker and Bianchi I cosmology with a cosmological constant
- A new geometric approach to Lie systems and physical applications
- Dynamical Studies of Equations from the Gambier Family
- Phase Splitting for Periodic Lie Systems
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- Multisymplectic structures and invariant tensors for Lie systems
- A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
- Free subgroups of free products and combinatorial hypermaps
- Reduction and reconstruction of multisymplectic Lie systems
- Mixed superposition rules for Lie systems, compatible geometric structures, and applications