The equation for fluid flow in porous media is analyzed in this paper with the aid of Lie symmetry method (LSM) and invariant subspace method (ISM). Infinitesimal generators, the entire geometric fields of the vectors and the symmetry groups of the equation being considered are given. One-dimensional optimal systems of sub-algebra are reported with corresponding reduced nonlinear ordinary differential equations. By means of ISM, we determine the exact solutions and invariant subspaces (ISs) for the equation under consideration. Moreover, with the aid of the new theorem of conservation, we establish the conservation laws (CLs) for the governing equation. The construction of the conserved vectors reveals the integrability and existence of soliton solutions of the equation for fluid flow in porous media.

中文翻译：

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多孔介质流体流动方程的对称性分析、不变子空间和守恒定律

本文借助李对称法（LSM）和不变子空间法（ISM）对多孔介质中的流体流动方程进行了分析。给出了无穷小生成器、向量的整个几何场和所考虑的方程的对称群。用相应的简化非线性常微分方程报告了子代数的一维最优系统。通过 ISM，我们确定了所考虑方程的精确解和不变子空间 (IS)。此外，借助新的守恒定理，我们建立了控制方程的守恒定律（CLs）。守恒向量的构造揭示了多孔介质中流体流动方程孤子解的可积性和存在性。