Superposition rules, Lie theorem and partial differential equations

Abstract

A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of Lie's Theorem for the case of systems of partial differential equations.

22 pages

References in corpus (1)

• Group theoretical approach to the intertwined Hamiltonians

Cited by in corpus (10)

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• The time-dependent quantum harmonic oscillator revisited: Applications to Quantum Field Theory
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• Quasi-Lie schemes and Emden--Fowler equations
• Nonlinear superpositions and Ermakov systems
• Integrability of Lie systems and some of its applications in physics
• A geometric approach to time evolution operators of Lie quantum systems
• Lie's Reduction Method and Differential Galois Theory in the Complex Analytic Context
• Superposition rules and stochastic Lie-Scheffers systems