标题： 叠加规则，李定理和偏微分方程 显示英文标题

标题： Superposition rules, Lie theorem and partial differential equations

摘要： 给出了一种严格的几何证明，证明了关于非自治常微分方程解的非线性叠加规则的李定理，填补了现有文献中存在的所有空白。 该证明基于叠加规则的一种替代但等价的定义：将其视为具有某些适当性质的叶状结构。 解决了叠加函数的唯一性问题，关键点在于从给定的向量场李代数构造出的叶状结构的余维数。 最后，作为支持使用这种替代的叠加规则定义的更有说服力的论据，展示了这种定义允许对偏微分方程组的情况立即推广李定理。 显示英文摘要

摘要： 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.