Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Superposition rules, Lie theorem and partial differential equations   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.

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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys