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 (4)

• Group theoretical approach to the intertwined Hamiltonians
• Riccati equation, Factorization Method and Shape Invariance
• The nonlinear superposition principle and the Wei-Norman method
• Integrability of Riccati equation from a group theoretical viewpoint

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