标题： 膜世界引力中包含体标量场的宇宙场方程的李对称性 显示英文标题

标题： Lie symmetries for the cosmological field equations in brane-world gravity with bulk scalar field

摘要： 我们研究了在膜宇宙学背景下引力场方程的群分类问题，考虑了体空间标量场的存在。 我们的研究围绕着一个五维时空展开，四维弗里德曼-勒梅特-罗伯逊-沃尔克几何嵌入其中。 此外，我们假设该标量场存在于这个五维几何（体空间）中，并且具有非零质量。 得到的场方程构成了一组非线性偏微分方程。 我们应用李对称条件来确定标量场势能的所有函数形式，确保场方程在单参数点变换下保持不变。 因此，我们发现只有指数势能表现出李对称性。 最后，利用李不变量构造相似变换，使我们能够推导出该系统的精确解。 显示英文摘要

摘要： We address the group classification problem for gravitational field equations within the context of brane-world cosmology, considering the presence of a bulk scalar field. Our investigation revolves around a five-dimensional spacetime, with the four-dimensional Friedmann--Lema\^{\i}tre--Robertson--Walker geometry embedded within it. Additionally, we assume that the scalar field exists in this five-dimensional geometry (bulk) and possesses a nonzero mass. The resulting field equations constitute a system of nonlinear partial differential equations. We apply the Lie symmetry condition to identify all functional forms of the scalar field potential, ensuring that the field equations remain invariant under one-parameter point transformations. Consequently, we find that only the exponential potential exhibits Lie symmetries. Finally, the Lie invariants are used to construct similarity transformations, which enable us to derive exact solutions for the system.